Modular Representation Theory of Finite Groupslassueur/en/teaching/... · 2021. 3. 24. · Modular Representation Theory of Finite Groups Jun.-Prof. Dr. Caroline Lassueur TU Kaiserslautern

Modular Representation Theory of Finite Groups

Jun.-Prof. Dr. Caroline LassueurTU Kaiserslautern

Skript zur Vorlesung, WS 2020/21(Vorlesung: 4SWS // Übungen: 2SWS)

Version: 23. August 2021

Contents

Foreword iii

Conventions iv

Chapter 1. Foundations of Representation Theory 61 (Ir)Reducibility and (in)decomposability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Schur’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Composition series and the Jordan-Hölder Theorem . . . . . . . . . . . . . . . . . . . . . . 84 The Jacobson radical and Nakayama’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 105 Indecomposability and the Krull-Schmidt Theorem . . . . . . . . . . . . . . . . . . . . . . 11

Chapter 2. The Structure of Semisimple Algebras 156 Semisimplicity of rings and modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 The Artin-Wedderburn structure theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 Semisimple algebras and their simple modules . . . . . . . . . . . . . . . . . . . . . . . . 22

Chapter 3. Representation Theory of Finite Groups 269 Linear representations of finite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2610 The group algebra and its modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2911 Semisimplicity and Maschke’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312 Simple modules over splitting fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Chapter 4. Operations on Groups and Modules 3613 Tensors, Hom’s and duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3614 Fixed and cofixed points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3915 Inflation, restriction and induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Chapter 5. The Mackey Formula and Clifford Theory 4516 Double cosets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4517 The Mackey formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4718 Clifford theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

i

Skript zur Vorlesung: Modular Representation Theory WS 2020/21 ii

Chapter 6. Projective Modules over the Group Algebra 5119 Radical, socle, head . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5120 Projective modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5421 Projective modules for the group algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5422 The Cartan matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5823 Symmetry of the group algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5924 Representations of cyclic groups in positive characteristic . . . . . . . . . . . . . . . . . . 62

Chapter 7. Indecomposable Modules 6425 Relative projectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6426 Vertices and sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7127 The Green correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7428 p-permutation modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7729 Green’s indecomposability theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Chapter 8. p-Modular Systems 8230 Complete discrete valuation rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8231 Splitting p-modular systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8532 Lifting idempotents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8833 Brauer Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Chapter 9. Brauer Characters 9334 Brauer characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9435 Back to decomposition matrices of finite groups . . . . . . . . . . . . . . . . . . . . . . . . 98

Chapter 10. Blocks 10436 The blocks of a ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10437 p-Blocks of finite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10638 Defect Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10839 Brauer’s 1st and 2nd Main Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Appendix 1: Background Material Module Theory 1000A Modules, submodules, morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1000B Free modules and projective modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003C Direct products and direct sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005D Exact sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006E Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1008F Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1011

Appendix 2: The Language of Category Theory 1013G Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013H Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016

Index of Notation 1018

Foreword

This text constitutes a faithful transcript of the lecture Modular Representation Theory held at theTU Kaiserslautern during the Winter Semester 2020/21 (14 Weeks, 4SWS Lecture + 2SWS Exercises).

Together with the necessary theoretical foundations the main aims of this lecture are to:

‚ provide students with a modern approach to finite group theory;

‚ learn about the representation theory of finite-dimensional algebras and in particular of thegroup algebra of a finite group;

‚ establish connections between the representation theory of a finite group over a field of positivecharacteristic and that over a field of characteristic zero;

‚ consistently work with universal properties and get acquainted with the language of categorytheory.

We assume as pre-requisites bachelor-level algebra courses dealing with linear algebra and elementarygroup theory, such as the standard lectures Grundlagen der Mathematik, Algebraische Strukturen, andEinführung in die Algebra. It is also strongly recommended to have attended the lectures CommutativeAlgebra and Character Theory of Finite Groups prior to this lecture. Therefore, in order to complementthese pre-requisites, but avoid repetitions, the Appendix deals formally with some background materialon module theory, but proofs are omitted.The main results of the lecture Character Theory of Finite Groups will be recovered through a differentand more general approach, thus it is formally not necessary to have attended this lecture already, butit definitely brings you some intuition.

Acknowledgement: I am grateful to Gunter Malle who provided me with the Skript of his lecture"Darstellungstheorie" hold at the TU Kaiserslautern in the WS 12/13, 13/14, 15/16 and 16/17, whichI used as a basis for the development of this lecture. I am also grateful to Niamh Farrell, who sharedwith me her text from 2019/20 to the second part of the lecture, which I had never taken myself priorto the WS20/21.I am also grateful to Kathrin Kaiser, Helena Petri and Bernhard Böhmler who mentioned typos to mein the preliminary version of these notes. Further comments, corrections and suggestions are of coursemore than welcome.

Conventions

Unless otherwise stated, throughout these notes we make the following general assumptions:

¨ all groups considered are finite;

¨ all rings considered are associative and unital (i.e. possess a neutral element for themultiplication, denoted 1);

¨ all modules considered are left modules;

¨ if K is a commutative ring and G a finite group, then all KG-modules considered areassumed to be free of finite rank when regarded as K -modules.

Skript zur Vorlesung: Modular Representation Theory WS 2020/21 v

Chapter 1. Foundations of Representation Theory

In this chapter we review four important module-theoretic theorems, which lie at the foundations ofrepresentation theory of finite groups:

1. Schur’s Lemma: about homomorphisms between simple modules.

2. The Jordan-Hölder Theorem: about "uniqueness" properties of composition series.

3. Nakayama’s Lemma: about an essential property of the Jacobson radical.

4. The Krull-Schmidt Theorem: about direct sum decompositions into indecomposable submodules.

Notation: throughout this chapter, unless otherwise specified, we let R denote an arbitrary unital andassociative ring.

References:

[Ben98] D. J. Benson. Representations and cohomology. I. Vol. 30. Cambridge Studies in AdvancedMathematics. Cambridge University Press, Cambridge, 1998.

[CR90] C. W. Curtis and I. Reiner. Methods of representation theory. Vol. I. John Wiley & Sons, Inc.,New York, 1990.

[Dor72] L. Dornhoff. Group representation theory. Part B: Modular representation theory. MarcelDekker, Inc., New York, 1972.

[NT89] H. Nagao and Y. Tsushima. Representations of finite groups. Academic Press, Inc., Boston,MA, 1989.

[Rot10] J. J. Rotman. Advanced modern algebra. 2nd ed. Providence, RI: American MathematicalSociety (AMS), 2010.

[Web16] P. Webb. A course in finite group representation theory. Vol. 161. Cambridge Studies inAdvanced Mathematics. Cambridge University Press, Cambridge, 2016.

1 (Ir)Reducibility and (in)decomposabilitySubmodules and direct sums of modules allow us to introduce the two main notions that will enable usto break modules in elementary pieces in order to simplify their study: simplicity and indecomposability.

6

Skript zur Vorlesung: Modular Representation Theory WS 2020/21 7

Definition 1.1 (simple/irreducible module / indecomposable module / semisimple module)

(a) An R-module M is called reducible if it admits an R-submodule U such that 0 Ĺ U Ĺ M .An R-module M is called simple, or irreducible, if it is non-zero and not reducible.

(b) An R-module M is called decomposable if M possesses two non-zero proper submodulesM1,M2 such that M “ M1 ‘M2. An R-module M is called indecomposable if it is non-zeroand not decomposable.

(c) An R-module M is called completely reducible or semisimple if it admits a direct sumdecomposition into simple R-submodules.

Our primary goal in Chapter 1 and Chapter 2 is to investigate each of these three concepts in details.

Remark 1.2Clearly any simple module is also indecomposable, resp. semisimple. However, the converse doesnot hold in general.

Exercise 1.3Prove that if pR,`, ¨q is a ring, then R˝ :“ R itself maybe seen as an R-module via left multiplicationin R , i.e. where the external composition law is given by

R ˆ R˝ ÝÑ R˝, pr, mq ÞÑ r ¨m .

We call R˝ the regular R-module.Prove that:

(a) the R-submodules of R˝ are precisely the left ideals of R ;

(b) I C R is a maximal left ideal of R ô R˝{I is a simple R-module, and I C R is a minimal leftideal of R ô I is simple when regarded as an R-submodule of R˝.

2 Schur’s LemmaSchur’s Lemma is a basic result, which lets us understand homomorphisms between simple modules,and, more importantly, endomorphisms of such modules.

Theorem 2.1 (Schur’s Lemma)

(a) Let V ,W be simple R-modules. Then:

(i) EndRpV q is a skew-field, and(ii) if V fl W , then HomRpV ,W q “ 0.

(b) If K is an algebraically closed field, A is a K -algebra, and V is a simple A-module such thatdimK V ă 8, then

EndApV q “ tλ IdV | λ P K u – K .


Proof :

(a) First, we claim that every f P HomRpV ,W qzt0u admits an inverse in HomRpW,V q.Indeed, f ‰ 0 ùñ ker f Ĺ V is a proper R-submodule of V and t0u ‰ Im f is a non-zero R-submodule of W . But then, on the one hand, ker f “ t0u, because V is simple, hence f is injective,and on the other hand, Im f “ W because W is simple. It follows that f is also surjective, hencebijective. Therefore, by Example A.4(d), f is invertible with inverse f´1 P HomRpW,V q.Now, (ii) is straightforward from the above. For (i), first recall that EndRpV q is a ring, which isobviously non-zero as EndRpV q Q IdV and IdV ‰ 0 because V ‰ 0 since it is simple. Thus, as anyf P EndRpV qzt0u is invertible, EndRpV q is a skew-field.

(b) Let f P EndApV q. By the assumptions on K , f has an eigenvalue λ P K . Let v P V zt0u be aneigenvector of f for λ. Then pf ´ λ IdV qpvq “ 0. Therefore, f ´ λ IdV is not invertible and

f ´ λ IdV P EndApV qpaqùñ f ´ λ IdV “ 0 ùñ f “ λ IdV .

Hence EndApV q Ď tλ IdV | λ P K u, but the reverse inclusion also obviously holds, so that

EndApV q “ tλ IdV u – K .

3 Composition series and the Jordan-Hölder TheoremFrom Chapter 2 on, we will assume that all modules we work with can be broken into simple modulesin the sense of the following definition.

Definition 3.1 (Composition series / composition factors / composition length)Let M be an R-module.

(a) A series (or filtration) of M is a finite chain of submodules

0 “ M0 Ď M1 Ď . . . Ď Mn “ M pn P Zě0q .

(b) A composition series of M is a series

0 “ M0 Ď M1 Ď . . . Ď Mn “ M pn P Zě0q

where Mi{Mi´1 is simple for each 1 ď i ď n. The quotient modules Mi{Mi´1 are called thecomposition factors (or the constituents) of M and the integer n is called the compositionlength of M .

Notice that, clearly, in a composition series all inclusions are in fact strict because the quotient modulesare required to be simple, hence non-zero.

Next we see that the existence of a composition series implies that the module is finitely generated.However, the converse does not hold in general. This is explained through the fact that the existenceof a composition series is equivalent to the fact that the module is both Noetherian and Artinian.


Definition 3.2 (Chain conditions / Artinian and Noetherian rings and modules)

(a) An R-module M is said to satisfy the descending chain condition (D.C.C.) on submodules(or to be Artinian) if every descending chain M “ M0 Ě M1 Ě . . . Ě Mr Ě . . . Ě t0u ofsubmodules eventually becomes stationary, i.e. D m0 such that Mm “ Mm0 for every m ě m0.

(b) An R-module M is said to satisfy the ascending chain condition (A.C.C.) on submodules (or tobe Noetherian) if every ascending chain 0 “ M0 Ď M1 Ď . . . Ď Mr Ď . . . Ď M of submoduleseventually becomes stationary, i.e. D m0 such that Mm “ Mm0 for every m ě m0.

(c) The ring R is called left Artinian (resp. left Noetherian) if the regular module R˝ is Artinian(resp. Noetherian).

Theorem 3.3 (Jordan-Hölder )Any series of R-submodules 0 “ M0 Ď M1 Ď . . . Ď Mr “ M (r P Zě0) of an R-module M may berefined to a composition series of M . In addition, if

0 “ M0 Ĺ M1 Ĺ . . . Ĺ Mn “ M pn P Zě0q

and0 “ M 1

0 Ĺ M 11 Ĺ . . . Ĺ M 1

m “ M pm P Zě0q

are two composition series of M , then m “ n and there exists a permutation π P Sn such thatM 1i{M 1

i´1 – Mπpiq{Mπpiq´1 for every 1 ď i ď n. In particular, the composition length is well-defined.

Proof : See Commutative Algebra.

Corollary 3.4If M is an R-module, then TFAE:

(a) M has a composition series;

(b) M satisfies D.C.C. and A.C.C. on submodules;

(c) M satisfies D.C.C. on submodules and every submodule of M is finitely generated.


Theorem 3.5 (Hopkins’ Theorem)If M is a module over a left Artinian ring, then TFAE:

(a) M has a composition series;

(b) M satisfies D.C.C. on submodules;

(c) M satisfies A.C.C. on submodules;

(d) M is finitely generated.

Proof : Without proof.


4 The Jacobson radical and Nakayama’s LemmaThe Jacobson radical is one of the most important two-sided ideals of a ring. As we will see in the nextsections and Chapter 2, this ideal carries a lot of information about the structure of a ring and that ofits modules.

Proposition-Definition 4.1 (Annihilator / Jacobson radical)

(a) Let M be an R-module. Then annRpMq :“ tr P R | rm “ 0 @ m P Mu is a two-sided idealof R , called annihilator of M .

(b) The Jacobson radical of R is the two-sided ideal

JpRq :“č

V simpleR-module

annRpV q “ tx P R | 1´ axb P Rˆ @ a, b P Ru .

(c) If V is a simple R-module, then there exists a maximal left ideal I C R such that V – R˝{I(as R-modules) and

JpRq “č

ICR,I maximalleft ideal

I .


Exercise 4.2

(a) Prove that any simple R-module may be seen as a simple R{JpRq-module.

(b) Conversely, prove that any simple R{JpRq-module may be seen as a simple R-module.[Hint: use a change of the base ring via the canonical morphism R ÝÑ R{JpRq.]

(c) Deduce that R and R{JpRq have the same simple modules.

Theorem 4.3 (Nakayama’s Lemma)If M is a finitely generated R-module and JpRqM “ M , then M “ 0.


Remark 4.4One often needs to apply Nakayama’s Lemma to a finitely generated quotient module M{U , whereU is an R-submodule of M . In that case the result may be restated as follows:

M “ U ` JpRqM ùñ U “ M


5 Indecomposability and the Krull-Schmidt TheoremWe now consider the notion of indecomposability in more details. Our first aim is to prove that inde-composability can be recognised at the endomorphism algebra of a module.

Definition 5.1A ring R is said to be local :ðñ RzRˆ is a two-sided ideal of R .

Example 1

(a) Any field K is local because K zKˆ “ t0u by definition.

(b) Exercise: Let p be a prime number and R :“ tab P Q | p - bu. Prove that RzRˆ “ tab P R | p|auand deduce that R is local.

(c) Exercise: Let K be a field and let R :“!

A “

¨

˝

a1 a2 ... an0 a1 ... an´1... . . . ...0 0 ... a1

˛

‚ P MnpK q)

. Prove that

RzRˆ “ tA P R | a1 “ 0u and deduce that R is local.

Proposition 5.2Let R be a ring. Then TFAE:

(a) R is local;

(b) RzRˆ “ JpRq, i.e. JpRq is the unique maximal left ideal of R ;

(c) R{JpRq is a skew-field.

Proof : Set N :“ RzRˆ.

(a)ñ(b): Clear: I C R proper left ideal ñ I Ď N . Hence, by Proposition-Definition 4.1(c),

JpRq “č


I Ď N .

Now, by (a) N is an ideal of R , hence N must be a maximal left ideal, even the unique one. Itfollows that N “ JpRq.

(b)ñ(c): If JpRq is the unique maximal left ideal of R , then in particular R ‰ 0 and R{JpRq ‰ 0. So letr P RzJpRq pbq“ Rˆ. Then obviously r ` JpRq P pR{JpRqqˆ. It follows that R{JpRq is a skew-field.

(c)ñ(a): Since R{JpRq is a skew-field by (c), R{JpRq ‰ 0, so that R ‰ 0 and there exists a P RzJpRq.Moreover, again by (c), a` JpRq P pR{JpRqqˆ, so that Db P RzJpRq such that

ab` JpRq “ 1` JpRq P R{JpRq

Therefore, D c P JpRq such that ab “ 1ć, which is invertible in R by Proposition-Definition 4.1(b).Hence Dd P R such that abd “ p1´ cqd “ 1 ñ a P Rˆ. Therefore RzJpRq “ Rˆ, and it followsthat RzRˆ “ JpRq which is a two-sided ideal of R .


Proposition 5.3 (Fitting’s Lemma)Let M be an R-module which has a composition series and let φ P EndRpMq be an endomorphismof M . Then there exists n P Zą0 such that

(i) φnpMq “ φnìpMq for every i ě 1;

(ii) kerpφnq “ kerpφnìq for every i ě 1; and

(iii) M “ φnpMq ‘ kerpφnq .

Proof : By Corollary 3.4 the module M satisfies both A.C.C. and D.C.C. on submodules. Hence the two chainsof submodules

φpMq Ě φ2pMq Ě . . . ,

kerpφq Ď kerpφ2q Ď . . .

eventually become stationary. Therefore we can find an index n satisfying both (i) and (ii).Exercise: Prove that M “ φnpMq ‘ kerpφnq.

Proposition 5.4Let M be an R-module which has a composition series. Then:

M is indecomposable ðñ EndRpMq is a local ring.

Proof : “ñ”: Assume that M is indecomposable. Let φ P EndRpMq. Then by Fitting’s Lemma there existsn P Zą0 such that M “ φnpMq ‘ kerpφnq. As M is indecomposable either φnpMq “ M andkerpφnq “ 0 or φnpMq “ 0 and kerpφnq “ M .¨ In the first case φ is bijective, hence invertible.¨ In the second case φ is nilpotent.

Therefore, N :“ EndRpMqzEndRpMqˆ “ tnilpotent elements of EndRpMqu.Claim: N is a two-sided ideal of EndRpMq.Let φ P N and m P Zą0 minimal such that φm “ 0. Then

φm´1pφρq “ 0 “ pρφqφm´1 @ ρ P EndRpMq .

As φm´1 ‰ 0, φρ and ρφ cannot be invertible, hence φρ, ρφ P N .Next let φ, ρ P N . If φ ` ρ “: ψ were invertible in EndRpMq, then by the previous argument wewould have ψ´1ρ, ψ´1φ P N , which would be nilpotent. Hence

ψ´1φ “ ψ´1pψ ´ ρq “ IdM ´ψ´1ρ

would be invertible.(Indeed, ψ´1ρ nilpotent ñ pIdM ´ψ´1ρqpIdM `ψ´1ρ` pψ´1ρq2 ` ¨ ¨ ¨ ` pψ´1ρqa´1q “ IdM , wherea is minimal such that pψ´1ρqa “ 0.)This is a contradiction. Therefore φ ` ρ P N , which proves that N is an ideal.Finally, it follows from the Claim and the definition that EndRpMq is local.

“ð”: Assume M is decomposable and let M1,M2 be proper submodules such that M “ M1 ‘M2. Thenconsider the two projections

π1 : M1 ‘M2 ÝÑ M1 ‘M2, pm1, m2q ÞÑ pm1, 0q


onto M1 along M2 and

π2 : M1 ‘M2 ÝÑ M1 ‘M2, pm1, m2q ÞÑ p0, m2q

onto M2 along M1. Clearly π1, π2 P EndRpMq but π1, π2 R EndRpMqˆ since they are not surjectiveby construction. Now, as π2 “ IdM ´π1 is not invertible it follows from the characterisation of theJacobson radical of Proposition-Definition 4.1(b) that π1 R JpEndRpMqq. Therefore

EndRpMqzEndRpMqˆ ‰ J pEndRpMqq

and it follows from Proposition 5.2 that EndRpMq is not a local ring.

Next, we want to be able to decompose R-modules into direct sums of indecomposable submodules. TheKrull-Schmidt Theorem will then provide us with certain uniqueness properties of such decompositions.

Proposition 5.5Let M be an R-module. If M satisfies either A.C.C. or D.C.C., then M admits a decomposition intoa direct sum of finitely many indecomposable R-submodules.

Proof : Let us assume that M is not expressible as a finite direct sum of indecomposable submodules. Thenin particular M is decomposable, so that we may write M “ M1 ‘W1 as a direct sum of two propersubmodules. W.l.o.g. we may assume that the statement is also false for W1. Then we also have adecomposition W1 “ M2‘W2, where M2 and W2 are proper sumbodules of W1 with the statement beingfalse for W2. Iterating this argument yields the following infinite chains of submodules:

W1 Ľ W2 Ľ W3 Ľ ¨ ¨ ¨ ,

M1 Ĺ M1 ‘M2 Ĺ M1 ‘M2 ‘M3 Ĺ ¨ ¨ ¨ .

The first chain contradicts D.C.C. and the second chain contradicts A.C.C.. The claim follows.

Theorem 5.6 (Krull–Schmidt)Let M be an R-module which has a composition series. If

M “ M1 ‘ ¨ ¨ ¨ ‘Mn “ M 11 ‘ ¨ ¨ ¨ ‘M 1

n1 pn, n1 P Zą0q

are two decomposition of M into direct sums of finitely many indecomposable R-submodules, thenn “ n1, and there exists a permutation π P Sn such that Mi – M 1

πpiq for each 1 ď i ď n and

M “ M 1πp1q ‘ ¨ ¨ ¨ ‘M

1πprq ‘

nà

j“r`1Mj for every 1 ď r ď n.

Proof : For each 1 ď i ď n let

πi : M “ M1 ‘ ¨ ¨ ¨ ‘Mn Ñ Mi, m1 ` . . .`mn ÞÑ mi

be the projection on the i-th factor of first decomposition, and for each 1 ď j ď n1 let

ψj : M “ M 11 ‘ ¨ ¨ ¨ ‘M 1

n1 Ñ M 1j , m11 ` . . .`m1n1 ÞÑ m1j

be the projection on the j-th factor of second decomposition.


Claim: if ψ P EndRpMq is such that π1 ˝ ψ|M1: M1 Ñ M1 is an isomorphism, then

M “ ψpM1q ‘M2 ‘ ¨ ¨ ¨ ‘Mn and ψpM1q – M1 .

Indeed : By the assumption of the claim, both ψ|M1: M1 Ñ ψpM1q and π1|ψpM1q : ψpM1q Ñ M1 must be

isomorphisms. Therefore ψpM1q X kerpπ1q “ 0, and for every m P M there exists m11 P ψpM1q such thatπ1pmq “ π1pm11q, hence m´m11 P kerpπ1q. It follows that

M “ ψpM1q ` kerpπ1q “ ψpM1q ‘ kerpπ1q “ ψpM1q ‘M2 ‘ ¨ ¨ ¨ ‘Mn .

Hence the Claim holds.Now, we have IdM “

řn1j“1 ψj , and so IdM1 “

řn1j“1 π1 ˝ ψj |M1

P EndRpM1q. But as M has a compositionseries, so has M1, and therefore EndRpM1q is local by Proposition 5.4. Thus if all the π1 ˝ ψj |M1

P

EndRpM1q are not invertible, they are all nilpotent and then so is IdM1 , which is in turn not invertible.This is not possible, hence it follows that there exists an index j such that

π1 ˝ ψj |M1: M1 Ñ M1

is an isomorphism and the Claim implies that M “ ψjpM1q ‘M2 ‘ ¨ ¨ ¨ ‘Mn and ψjpM1q – M1.We then set πp1q :“ j . By definition ψjpM1q Ď M 1

j as M 1j is indecomposbale, so that

ψjpM1q – M 1j “ M 1

πp1q .

Finally, an induction argument (Exercise!) yields:

M “ M 1πp1q ‘ ¨ ¨ ¨ ‘M 1

πprq ‘nà

j“r`1Mj ,

mit M 1πpiq – Mi (1 ď i ď r). In particular, the case r “ n implies the equality n “ n1.

Chapter 2. The Structure of Semisimple Algebras

In this chapter we study an important class of rings: the class of rings R which are such that any R-module can be expressed as a direct sum of simple R-submodules. We study the structure of such ringsthrough a series of results essentially due to Artin and Wedderburn. At the end of the chapter we willassume that the ring is a finite dimension algebra over a field and start the study of its representationtheory.

Notation: throughout this chapter, unless otherwise specified, we let R denote a unital and associativering.

References:





6 Semisimplicity of rings and modulesTo begin with, we prove three equivalent characterisations for the notion of semisimplicity.

Proposition 6.1If M is an R-module, then the following assertions are equivalent:

(a) M is semisimple, i.e. M “ ‘iPISi for some family tSiuiPI of simple R-submodules of M;

(b) M “ř

iPI Si for some family tSiuiPI of simple R-submodules of M;

(c) every R-submodule M1 Ď M admits a complement in M , i.e. D an R-submodule M2 Ď Msuch that M “ M1 ‘M2.

15


Proof :(a)ñ(b): is trivial.

(b)ñ(c): Write M “ř

iPI Si, where Si is a simple R-submodule of M for each i P I . Let M1 Ď M be anR-submodule of M . Then consider the family, partially ordered by inclusion, of all subsets J Ď Isuch that(1)

ř

iPJ Si is a direct sum, and(2) M1 X

ř

iPJ Si “ 0.Clearly this family is non-empty since it contains the empty set. Thus Zorn’s Lemma yields theexistence of a maximal element J0. Now, set

M 1 :“ M1 `ÿ

iPJ0

Si “ M1 ‘ÿ

iPJ0

Si ,

where the second equality holds by (1) and (2). Therefore, it suffices to prove that M “ M 1, i.e.that Si Ď M 1 for every i P I . But if j P I is such that Sj Ę M 1, the simplicity of Sj implies thatSj XM 1 “ 0 and it follows that

M 1 ` Sj “ M1 ‘

˜

ÿ

iPJ0

Si

¸

‘ Sj

in contradiction with the maximality of J0. The claim follows.

(b)ñ(a): follows from the argument above with M1 “ 0.

(c)ñ(b): Let M1 be the sum of all simple R-submodules in M . By (c) there exists a complement M2 Ď Mto M1, i.e. such that M “ M1 ‘M2. If M2 “ 0, we are done. If M2 ‰ 0, then M2 must contain asimple R-submodule (Exercise: prove this fact), say N . But then N Ď M1 by definition of M1, acontradiction. Thus M2 “ 0 and so M “ M1.

Example 2

(a) The zero module is completely reducible, but neither reducible nor irreducible!

(b) If S1, . . . , Sn are simple R-modules, then their direct sum S1‘ . . .‘Sn is completely reducibleby definition.

(c) The following exercise shows that there exists modules which are not completely reducible.

Exercise (Sheet 1): Let K be a field and let A be the K -algebra ` a1 a

0 a1

˘

| a1, a P K(

. Con-sider the A-module V :“ K 2, where A acts by left matrix multiplication. Prove that:

(1) tp x0 q | x P K u is a simple A-submodule of V ; but(2) V is not semisimple.

(d) Exercise (Sheet 1): Prove that any submodule and any quotient of a completely reduciblemodule is again completely reducible.


Theorem-Definition 6.2 (Semisimple ring)A ring R satisfying the following equivalent conditions is called semisimple.

(a) All short exact sequences of R-modules split.

(b) All R-modules are semisimple.

(c) All finitely generated R-modules are semisimple.

(d) The regular left R-module R˝ is semisimple, and is a direct sum of a finite number of minimalleft ideals.

Proof : First, (a) and (b) are equivalent as a consequence of Lemma D.4 and the characterisation of semisimplemodules given by Proposition 6.1(c). The implication (b) ñ (c) is trivial, and it is also trivial that (c)implies the first claim of (d), which in turn implies the second claim of (d). Indeed, if R˝ “

À

iPI Li forsome family tLiuiPI of minimal left ideals. Then, by definition of a direct sum, there exists a finite numberof indices i1, . . . , in P I such that 1R “ xi1 ` . . . ` xin with xij P Lij for each 1 ď j ď n. Therefore eacha P R may be expressed in the form

a “ a ¨ 1R “ axi1 ` . . .` axin

and hence R˝ “ Li1 ` . . .` Lin .Therefore, it remains to prove that (d) ñ (b). So, assume that R satisfies (d) and let M be an arbitrarynon-zero R-module. Then write M “

ř

mPM R ¨m. Now, each cyclic submodule R ¨m of M is isomorphicto an R-submodule of R˝, which is semisimple by (d). Thus R ¨m is semisimple as well by Example 2(d).Finally, it follows from Proposition 6.1(b) that M is semisimple.

Example 3Fields are semisimple. Indeed, if V is a finite-dimensional vector space over a field K of dimension n,then choosing a K -basis te1, ¨ ¨ ¨ , enu of V yields V “ Ke1 ‘ . . . ‘ Ken, where dimK pKeiq “ 1,hence Kei is a simple K -module for each 1 ď i ď n. Hence, the claim follows from Theorem-Definition 6.2(c).

Corollary 6.3Let R be a semisimple ring. Then:

(a) R˝ has a composition series;

(b) R is both left Artinian and left Noetherian.

Proof :

(a) By Theorem-Definition 6.2(d) the regular module R˝ admits a direct sum decomposition into a finitenumber of minimal left ideals. Removing one ideal at a time, we obtain a composition series for R˝.

(b) Since R˝ has a composition series, it satisfies both D.C.C. and A.C.C. on submodules by Corol-lary 3.4. In other words, R is both left Artinian and left Noetherian.

Next, we show that semisimplicity is detected by the Jacobson radical. This leads us to introduce aslightly weaker concept: the notion of J-semisimplicity.


Definition 6.4 (J-semimplicity)A ring R is said to be J-semisimple if JpRq “ 0.

Exercise 6.5 (Sheet 1)Let R “ Z. Prove that JpZq “ 0, but not all Z-modules are semisimple. In other words, Z isJ-semisimple but not semisimple.

Proposition 6.6Any left Artinian ring R is J-semisimple if and only if it is semisimple.

Proof : “ñ”: Assume R ‰ 0 and R is not semisimple. Pick a minimal left ideal I0 Ĳ R (e.g. a minimalelement of the family of non-zero principal left ideals of R ). Then 0 ‰ I0 ‰ R since I0 seen as anR-module is simple.Claim: I0 is a direct summand of R˝.Indeed: since

I0 ‰ 0 “ JpRq “č


I

there exists a maximal left ideal m0 C R which does not contain I0. Thus I0 Xm0 “ t0u and so wemust have R˝ “ I0 ‘m0, as R{m0 is simple. Hence the Claim.Notice that then m0 ‰ 0, and pick a minimal left ideal I1 in m0. Then 0 ‰ I1 ‰ m0, else R wouldbe semisimple. The Claim applied to I1 yields that I1 is a direct summand of R˝, hence also in m0.Therefore, there exists a non-zero left ideal m1 such that m0 “ I1 ‘m1. Iterating this process, weobtain an infinite descending chain of ideals

m0 Ľ m1 Ľ m2 Ľ ¨ ¨ ¨

contradicting D.C.C.“ð”: Conversely, if R is semisimple, then R˝ – R{JpRq ‘ JpRq by Theorem-Definition 6.2 and so as

R-modules,JpRq “ JpRq ¨ pR{JpRq ‘ JpRqq “ JpRq ¨ JpRq

so that by Nakayama’s Lemma JpRq “ 0.

Proposition 6.7The quotient ring R{JpRq is J-semisimple.

Proof : Since by Exercise 4.2 the rings R and R :“ R{JpRq have the same simple modules (seen as abeliangroups), Proposition-Definition 4.1(a) yields:

JpRq “č

V simpleR-module

annRpV q “č

V simpleR-module

annRpV q ` JpRq “ JpRq{JpRq “ 0

7 The Artin-Wedderburn structure theoremThe next step in analysing semisimple rings and modules is to sort simple modules into isomorphismclasses. We aim at proving that each isomorphism type of simple modules actually occurs as a directsummand of the regular module. The first key result in this direction is the following proposition:


Proposition 7.1Let M be a semisimple R-module. Let tMiuiPI be a set of representatives of the isomorphism classesof simple R-submodules of M and for each i P I set

Hi :“ÿ

VĎMV–Mi

V .

Then the following statements hold:

(i) M –À

iPI Hi ;

(ii) every simple R-submodule of Hi is isomorphic to Mi ;

(iii) HomRpHi, Hi1q “ t0u if i ‰ i1; and

(iv) if M “À

jPJ Vj is an arbitrary decomposition of M into a direct sum of simple submodules,then

rHi :“ÿ

jPJVj–Mi

Vj “à

jPJVj–Mi

Vj “ Hi .

Proof : We shall prove several statements which, taken together, will establish the theorem.

Claim 1: If M “À

jPJ Vj as in (iv) and W is an arbitrary simple R-submodule of M , then D j P J suchthat W – Vj .Indeed: if tπj : M “

À

jPJ Vj ÝÑ VjujPJ denote the canonical projections on the j-th summand, thenD j P J such that πjpW q ‰ 0. Hence πj |W : W ÝÑ Vj is an R-isomorphism as both W and Vj are simple.

Claim 2: If M “À

jPJ Vj as in (iv), then M “À

iPIrHi and for each i P I , every simple R-submodule of

rHi is isomorphic to Mi.Indeed: the 1st statement of the claim is obvious and the 2nd statement follows from Claim 1 appliedto rHi.

Claim 3: If W is an arbitrary simple R-submodule of M , then there is a unique i P I such that W Ď rHi.Indeed: it is clear that there is a unique i P I such that W – Mi. Now consider w P W zt0u and writew “

ř

jPJ wj PÀ

jPJ Vj with wj P Vj . The proof of Claim 1 shows that if any summand wj ‰ 0, thenπjpW q ‰ 0, and hence W – Vj . Therefore wj “ 0 unless Vj – Mi, and hence w P rHi, so that W Ď rHi.

Claim 4: HomRprHi, rHi1q “ t0u if i ‰ i1.Indeed: if 0 ‰ f P HomRprHi, rHi1q and i ‰ i1, then there must exist a simple R-submodule W of rHi suchthat fpW q ‰ 0, hence as W is simple, f |W : W ÝÑ fpW q is an R-isomorphism. It follows from Claim 2,that fpW q is a simple R-submodule of rHi1 isomorphic to Mi. This contradicts Claim 2 saying that everysimple R-submodule of rHi1 is isomorphic to Mi1 fl Mi.

Now, it is clear that rHi Ď Hi by definition. On the other hand it follows from Claim 3, that Hi Ď rHi.Hence Hi “ rHi for each i P I , hence (iv). Then Claim 2 yields (i) and (ii), and Claim 4 yields (iii).

We give a name to the submodules tHiuiPI defined in Propostion 7.1:


Definition 7.2

(a) If M is a semisimple R-module and S is a simple R-module, then the S-homogeneous com-ponent of M , denoted SpMq, is the sum of all simple R-submodules of M isomorphic to S.

(b) We let MpRq denote a set of representatives of the isomorphism classes of simple R-modules.

Exercise 7.3 (Sheet 2)Let R be a semisimple ring. Prove the following statements.

(a) Every non-zero left ideal I of R is generated by an idempotent of R , in other words D e P Rsuch that e2 “ e and I “ Re. (Hint: choose a complement I 1 for I , so that R˝ “ I ‘ I 1 andwrite 1 “ e` e1 with e P I and e1 P I 1. Prove that I “ Re.)

(b) If I is a non-zero left ideal of R , then every morphism in HomRpI, R˝q is given by rightmultiplication with an element of R .

(c) If e P R is an idempotent, then EndRpReq – peReqop (the opposite ring) as rings via the mapf ÞÑ efpeqe. In particular EndRpR˝q – Rop via f ÞÑ fp1q.

(d) A left ideal Re generated by an idempotent e of R is minimal (i.e. simple as an R-module) ifand only if eRe is a division ring. (Hint: Use Schur’s Lemma.)

(e) Every simple left R-module is isomorphic to a minimal left ideal in R , i.e. a simple R-submodule of R˝.

We recall that:

Definition 7.4 (Centre)The centre of a ring pR,`, ¨q is Z pRq :“ ta P R | a ¨ x “ x ¨ a @ x P Ru.

Theorem 7.5 (Wedderburn)If R is a semisimple ring, then the following assertions hold.

(a) If S P MpRq, then SpR˝q ‰ 0. Furthermore, |MpRq| ă 8.

(b) We haveR˝ “

à

SPMpRqSpR˝q ,

where each homogenous component SpR˝q is a two-sided ideal of R and SpR˝qT pR˝q “ 0 ifS ‰ T P MpRq.

(c) Each SpR˝q is a simple left Artinian ring, the identity element of which is an idempotentelement of R lying in Z pRq.


Proof :

(a) By Exercise 7.3(e) every simple left R-module is isomorphic to a minimal left ideal of R , i.e. asimple submodule of R˝. Hence if S P MpRq, then SpR˝q ‰ 0. Now, by Theorem-Definition 6.2,the regular module admits a decomposition

R˝ “à

jPJVj

into a direct sum of a finite number of minimal left ideals Vj of R , and by Claim 1 in the proof ofProposition 7.1 any simple submodule of R˝ is isomorphic to Vj for some j P J . Hence |MpRq| ă 8.

(b) Proposition 7.1(iv) also yields SpR˝q “À

Vj–S Vj and Proposition 7.1(i) implies that

R˝ “à

SPMpRqSpR˝q .

Next notice that each homogeneous component is a left ideal of R , since it is by definition a sum ofleft ideals. Now let L be a minimal left ideal contained in SpR˝q, and let x P T pR˝q for a T P MpRqwith S ‰ T . Then Lx Ď T pR˝q and because φx : R˝ ÝÑ R˝, m ÞÑ mx is an R-endomorphismof R˝, then either Lx “ φxpLq is zero or it is again a minimal left ideal, isomorphic to L. However,as S ‰ T , we have Lx “ 0. Therefore SpR˝qT pR˝q “ 0, which implies that SpR˝q is also a rightideal, hence two-sided.

(c) Part (b) implies that the homogeneous components are rings. Then, using Exercise 7.3(a), we maywrite 1R “

ř

SPMpRq eS , where SpR˝q “ ReS with eS idempotent. Since SpR˝q is a two-sidedideal, in fact SpR˝q “ ReS “ eSR . It follows that eS is an identity element for SpR˝q.To see that eS is in the centre of R , consider an arbitrary element a P R and write a “

ř

TPMpRq aTwith aT P T pR˝q. Since SpR˝qT pR˝q “ 0 if S ‰ T P MpRq, we have eSeT “ δST . Thus, as eT isan identity element for the T -homogeneous component, we have

eSa “ eSÿ

TPMpRqaT “ eS

ÿ

TPMpRqeTaT “

ÿ

TPMpRqeSeTaT

“ eSaS“ aSeS“

ÿ

TPMpRqaTeTeS “ p

ÿ

TPMpRqaTeT qeS “ p

ÿ

TPMpRqaT qeS “ aeS .

Finally, if L ‰ 0 is a two-sided ideal in SpR˝q, then L must contain all the minimal left ideals ofR isomorphic to S as a consequence of Exercise 7.3 (check it!). It follows that L “ SpR˝q andtherefore SpR˝q is a simple ring. It is left Artinian, because it is semissimple as an R-module.

Scholium 7.6If R is a semisimple ring, then there exists a set of idempotent elements teS | S P MpRqu such that

(i) eS P Z pRq for each S P MpRq;

(ii) eSeT “ δSTeS for all S, T P MpRq;

(iii) 1R “ř

SPMpRq eS ;

(iv) R “À

SPMpRq ReS , where each ReS is a simple ring.

Idempotents satisfying Property (i) are called central idempotents, and idempotents satisfying Prop-erty (ii) are called orthogonal.


Remark 7.7Remember that if R is a semisimple ring, then the regular module R˝ admits a composition series.Therefore it follows from the Jordan-Hölder Theorem that

R˝ “à

SPMpRqSpR˝q –

à

SPMpRq

nSà

i“1S

for uniquely determined integers nS P Zą0.

Theorem 7.8 (Artin-Wedderburn)If R is a semisimple ring, then, as a ring,

R –ź

SPMpRqMnS pDSq ,

where DS :“ EndRpSqop is a division ring.

Before we proceed with the proof of the theorem, first recall that if we have a direct sum decompositionU “ U1 ‘ ¨ ¨ ¨ ‘ Ur (r P Zą0), then EndRpUq is isomorphic to the ring of r ˆ r-matrices in which thepi, jq entry lies in HomRpUj , Uiq. This is because any R-endomorphism φ : U ÝÑ U may be written asa matrix of components φ “ pφijq1ďi,jďr where φij : Uj

inc.ÝÑ U φ

ÝÑ U proj.ÝÑ Ui, and when viewed in this

way R-endomorphisms compose in the manner of matrix multiplication. (Known from the GDM-lectureif R is a field. The same holds over an arbitrary ring R .)

Proof : By Exercise 7.3(c), we haveEndRpR˝q – Rop

as rings. On the other hand, since HomRpSpR˝q, T pR˝qq “ 0 for S fl T (e.g. by Schur’s Lemma, or byProposition 7.1), the above observation yields

EndRpR˝q –ź

SPMpRqEndRpSpR˝qq

where EndRpSpR˝qq – MnS pEndRpSqq – MnS pEndRpSqopqop. Therefore, setting DS :“ EndRpSqop yieldsthe result. For by Schur’s Lemma EndRpSq is a division ring, hence so is the opposite ring.

8 Semisimple algebras and their simple modulesFrom now on we leave the theory of modules over arbitrary rings and focus on finite-dimensionalalgebras over a field K . Algebras are in particular rings, and since K -algebras and their modulesare in particular K -vector spaces, we may consider their dimensions to obtain further information. Inparticular, we immediately see that finite-dimensional K -algebras are necessarily left Artinian rings.Furthermore, the structure theorems of the previous section tell us that if A is a semisimple algebraover a field K , then

A˝ “à

SPMpAqSpA˝q –

à

SPMpAq

nSà

i“1S

where nS corresponds to the multiplicity of the isomorphism class of the simple module S as a directsummand of A˝ in any given decomposition of A˝ into a finite direct sum of simple submodules. We shall


see that over an algebraically closed field the number of simple A-modules is detected by the centreof A and also obtain information about the simple modules of algebras, which are not semisimple.

Exercise 8.1 (Sheet 2)Let A be an arbitrary K -algebra over a commutative ring K .

(a) Prove that Z pAq is a K -subalgebra of A.

(b) Prove that if K is a field and A ‰ 0, then K ÝÑ Z pAq, λ ÞÑ λ1A is an injective K -homomorphism.

(c) Prove that if A “ MnpK q, then Z pAq “ K In, i.e. the K -subalgebra of scalar matrices. (Hint:use the standard basis of MnpK q.)

(d) Assume A is the algebra of 2ˆ 2 upper-triangular matrices over K . Prove that

Z pAq “ ` a 0

0 a˘

| a P K(

.

We obtain the following Corollary to Wedderburn’s and Artin-Wedderburn’s Theorems:

Theorem 8.2Let A be a semisimple finite-dimensional algebra over an algebraically closed field K , and letS P MpAq be a simple A-module. Then the following statements hold:

(a) SpA˝q – MnS pK q and dimK pSpA˝qq “ n 2S ;

(b) dimK pSq “ nS ;

(c) dimK pAq “ř

SPMpAq dimK pSq2 ;

(d) |MpAq| “ dimK pZ pAqq.

Proof :

(a) Since K “ K , Schur’s Lemma implies that EndApSq – K . Hence the division ring DS in thestatement of the Artin-Wedderburn Theorem is DS “ EndApSqop – K op “ K . Hence Artin-Wedderburn (and its proof) applied to the case R “ SpA˝q yields SpA˝q – MnS pK q. HencedimK pSpA˝qq “ n 2

S .(b) Since SpA˝q is a direct sum of nS copies of S, (a) yields:

n2S “ nS ¨ dimK pSq ùñ dimK pSq “ nS

(c) follows directly from (a) and (b).(d) Since by Artin-Wedderburn and (a) we have A –

ś

SPMpAqMnS pK q, clearly

Z pAq –ź

SPMpAqZ pMnS pK qq “

ź

SPMpAqK InS ,

where dimK pK InS q “ 1. The claim follows.


Corollary 8.3Let A be a finite-dimensional algebra over an algebraically closed field K . Then the number ofsimple A-modules is equal to dimK pZ pA{JpAqqq.

Proof : We have observed that A and A{JpAq have the same simple modules (see Exercise 4.2), hence|MpAq| “ |MpA{JpAqq|. Moreover, the quotient A{JpAq is J-semisimple by Proposition 6.7, hencesemisimple by Proposition 6.6 because finite-dimensional algebras are left Artinian rings. Thereforeit follows from Theorem 8.2(d) that

|MpAq| “ |MpA{JpAqq| “ dimK`

Z pA{JpAqq˘

.

Corollary 8.4Let A be a finite-dimensional algebra over an algebraically closed field K . If A is commutative, thenany simple A-module has K -dimension 1.

Proof : First assume that A is semisimple. As A is commutative, A “ Z pAq. Hence parts (d) and (c) ofTheorem 8.2 yield

|MpAq| “ dimK pAq “ÿ

SPMpAqdimK pSq2loooomoooon

ě1

,

which forces dimK pSq “ 1 for each S P MpAq.Now, if A is not semissimple, then again we use the fact that A and A{JpAq have the same simple modules(that is seen as abelian groups). Because A{JpAq is semisimple and also commutative, the argumentabove tells us that all simple A{JpAq-modules have K -dimension 1. The claim follows.

Finally, we emphasise that in this section the assumption that the field K is algebraically closed is ingeneral too strong and that it is possible to weaken this hypothesis so that Theorem 8.2, Corollary 8.3and Corollary 8.4 still hold.

Indeed, if K “ K is algebraically closed, then Part (b) of Schur’s Lemma tells us that EndApSq – Kfor any simple A-module S. This is the crux of the proof of Theorem 8.2. The following terminologydescribes this situation.

Definition 8.5Let A be a finite-dimensional K -algebra. Then:

(a) A is called split if EndApSq – K for every simple A-module S; and

(b) an extension field K 1 of K is called a splitting field for A if the K 1-algebra K 1 bK A is split.

Of course if A is split then K itself is a splitting field for A.

Remark 8.6In fact for a finite-dimensional K -algebra A, the following assertions are equivalent:

(a) A is split;

(b) the product, for S running through MpAq, of the structural homomorphisms A ÝÑ EndK pSq


(mapping a P A to the K -linear map S ÝÑ S,m ÞÑ am) induces an isomorphism of K -algebras

A{JpAq –ź

SPMpAqEndK pSq .

This is a variation of the Artin-Wedderburn Theorem we have seen in the previous section.

Chapter 3. Representation Theory of Finite Groups

Representation theory of finite groups is originally concerned with the ways of writing a finite group Gas a group of matrices, that is using group homomorphisms from G to the general linear group GLnpK qof invertible nˆn-matrices with coefficients in a field K for some positive integer n. Thus, we shall firstdefine representations of groups using this approach. Our aim is then to translate such homomorphismsG ÝÑ GLnpK q into the language of module theory in order to be able to apply the theory we havedeveloped so far. In particular, our first aim is to understand what the general theory of semisimplerings and the Artin-Wedderburn theorem bring to the theory of representations of finite groups.

Notation: throughout this chapter, unless otherwise specified, we let G denote a finite group and K bea commutative ring. Moreover, in order to simplify some arguments, we assume that all KG-modulesconsidered are free of finite rank when regarded as K -modules. (This implies, in particular, that theyare finitely generated as KG-modules.)

References:

[Alp86] J. L. Alperin. Local representation theory. Vol. 11. Cambridge Studies in Advanced Mathe-matics. Cambridge University Press, Cambridge, 1986.




[LP10] K. Lux and H. Pahlings. Representations of groups. Vol. 124. Cambridge Studies in AdvancedMathematics. Cambridge University Press, Cambridge, 2010.


9 Linear representations of finite groupsTo begin with, we review elementary definitions and examples about representations of finite groups.

26


Definition 9.1 (K -representation, matrix representation)

(a) A K -representation of G is a group homomorphism ρ : G ÝÑ GLpV q, where V – K n

(n P Zą0) is a free K -module of finite rank and GLpV q :“ AutK pV q.

(b) A matrix representation of G over K is a group homomorphism X : G ÝÑ GLnpK q (n P Zą0).

In both cases the integer n is called the degree of the representation.

(c) If K is a field, then a K -representation (resp. a matrix representation) is called an ordinaryrepresentation if charpK q “ 0 (or more generally if charpK q - |G|), and it is called a modularrepresentation if charpK q | |G|.

Remark 9.2Both concepts of a representation and of a matrix representation are closely related. Indeed, recallthat every choice of a basis B of V yields a group isomorphism

αB : GLpV q ÝÑ GLnpK q, φ ÞÑ pφqB

(where pφqB denotes the matrix of φ w.r.t. the basis B). Thus, a K -representation ρ : G ÝÑ GLpV qtogether with the choice of a basis B of V gives rise to a matrix representation of G:

G GLpV q GLnpK qρ αB

Conversely, any matrix representation X : G ÝÑ GLnpK q gives rise to a K -representation

ρ : G ÝÑ GLpK nqg ÞÑ ρpgq : K n ÝÑ K n, v ÞÑ Xpgqv ,

namely we set V “ K n, see v as a column vector expressed in the standard basis of K n and Xpgqvdenotes the standard matrix multiplication.

Example 4

(a) If G is an arbitrary finite group, then

ρ : G ÝÑ GLpK q – Kˆg ÞÑ ρpgq :“ IdK Ø 1K

is a K -representation of G, called the trivial representation of G.

(b) If X is a finite G-set, i.e. a finite set endowed with a left action ¨ : G ˆ X ÝÑ X , and V is afree K -module with basis tex | x P Xu, then

ρX : G ÝÑ GLpV qg ÞÑ ρX pgq : V ÝÑ V , ex ÞÑ eg¨x

is a K -representation of G, called the permutation representation associated with X .

Two particularly interesting examples are the following:


(1) if G “ Sn (n ě 1) is the symmetric group on n letters and and X “ t1, 2, . . . , nu thenρX is called natural representation of Sn;

(2) if X “ G and the left action ¨ : G ˆ X ÝÑ X is just the multiplication in G, thenρX “: ρreg is called the regular representation of G.

Definition 9.3 (Homomorphism of representations, equivalent representations)Let ρ1 : G ÝÑ GLpV1q and ρ2 : G ÝÑ GLpV2q be two K -representations of G, where V1, V2 aretwo non-zero free K -modules of finite rank.

(a) A K -homomorphism α : V1 ÝÑ V2 such that ρ2pgq ˝ α “ α ˝ ρ1pgq for each g P G is called ahomomorphism of representations (or a G-homomorphism) between ρ1 and ρ2.

V1 V1

V2 V2

ρ1pgq

α ö α

ρ2pgq

(b) If, moreover, α is a K -isomorphism, then it is called an isomorphism of representations (or aG-isomorphism), and the K -representations ρ1 and ρ2 are called equivalent (or similar, orisomorphic). In this case we write ρ1 „ ρ2.

Remark 9.4

(a) Equivalent representations have the same degree.

(b) Clearly „ is an equivalence relation.

(c) In consequence, it essentially suffices to study representations up to equivalence (as it es-sentially suffices to study groups up to isomorphism).

Definition 9.5 (G-invariant subspace, irreducibility)Let ρ : G ÝÑ GLpV q be a K -representation of G.

(a) A K -submodule W Ď V is called G-invariant if

ρpgq`

W˘

Ď W @g P G .

(In fact in this case the reverse inclusion holds as well, since for each w P W we can writew “ ρpgg´1qpwq “ ρpgq

`

ρpg´1qpwq˘

P ρpgq`

W˘

, hence ρpgq`

W˘

“ W .)

(b) The representation ρ is called irreducible if it admits exactly two G-invariant K -submodules,namely 0 and V itself; else it is called reducible.

Notice that V itself and the zero K -module 0 are always G-invariant.


Definition 9.6 (Subrepresentation)If ρ : G ÝÑ GLpV q is a K -representation and 0 ‰ W Ď V is a G-invariant K -submodule, then

ρW : G ÝÑ GLpW qg ÞÑ ρW pgq :“ ρpgq|W : W ÝÑ W

is called a subrepresentation of ρ. (This is clearly again a K -representation of G.)

With this definition, it is clear that a representation ρ : G ÝÑ GLpV q is irreducible if and only if ρdoes not possess any proper subrepresentation.

10 The group algebra and its modulesWe now want to be able to see K -representations of a group G as modules, and more precisely asmodules over a K -algebra depending on the group G, which is called the group algebra:

Lemma-Definition 10.1 (Group algebra)The group ring KG is the ring whose elements are the K -linear combinations

ř

gPG λgg withλg P K @g P G, and addition and multiplication are given by

ÿ

gPGλgg`

ÿ

gPGµgg “

ÿ

gPGpλg ` µgqg and

`

ÿ

gPGλgg

˘

¨`

ÿ

hPGµhh

˘

“ÿ

g,hPGpλgµhqgh

respectively. Thus KG is a K -algebra, which as a K -module is free with basis G. Hence we usuallycall KG the group algebra of G over K rather than simply group ring.

Proof : By definition KG is a free K -module with basis G, and the multiplication in G is extended by K -bilinearity to the given multiplication ¨ : KGˆKG ÝÑ KG. It is then straightforward that KG bears boththe structures of a ring and of a K -module. Finally, axiom (A3) of K -algebras follows directly from thedefinition of the multiplication and the commutativity of K .

Remark 10.2Clearly:

¨ 1KG “ 1G ;

¨ the K -rank of KG is |G|;

¨ KG is commutative if and only if G is an abelian group;

¨ if K is a field, or more generally (left) Artinian, then KG is a left Artinian ring, so that byHopkin’s Theorem a KG-module is finitely generated if and only if it admits a compositionseries.

Also notice that since G is a group, the map KG ÝÑ KG defined by g ÞÑ g´1 for each g P G is ananti-automorphism. It follows that any left KG-module M may be regarded as a right KG-modulevia the right G-action m ¨ g :“ g´1 ¨m. Thus the sidedness of KG-modules is not usually an issue.


As KG is a K -algebra, we may of course consider modules over KG and we recall that any KG-moduleis in particular a K -module. Moreover, we adopt the following convention, which is automatically sat-isfied if K is a field.

Convention: in the sequel all KG-modules considered are assumed to be free of finite rank whenregarded as K -modules.

Proposition 10.3

(a) Any K -representation ρ : G ÝÑ GLpV q of G gives rise to a KG-module structure on V , wherethe external composition law is defined by the map

¨ : G ˆ V ÝÑ Vpg, vq ÞÑ g ¨ v :“ ρpgqpvq

extended by K -linearity to the whole of KG.

(b) Conversely, every KG-module pV ,`, ¨q defines a K -representation

ρV : G ÝÑ GLpV qg ÞÑ ρV pgq : V ÝÑ V , v ÞÑ ρV pgq :“ g ¨ v

of the group G.

Proof : (a) Since V is a K -module it is equipped with an internal addition ` such that pV ,`q is an abeliangroup. It is then straightforward to check that the given external composition law defined aboveverifies the KG-module axioms.

(b) A KG-module is in particular a K -module for the scalar multiplication defined for all λ P K and allv P V by

λv :“ p λ 1Gloomoon

PKG

q ¨ v .

Moreover, it follows from the KG-module axioms that ρV pgq P GLpV q and also that

ρV pg1g2q “ ρV pg1q ˝ ρV pg2q

for all g1, g2 P G, hence ρV is a group homomorphism.

Example 5Via Proposition 10.3 the trivial representation (Example 4(a)) corresponds to the so-called trivialKG-module, that is the commutative ring K itself seen as a KG-module via the G-action

¨ : G ˆ K ÝÑ Kpg, λq ÞÝÑ g ¨ λ :“ λ

extended by K -linearity to the whole of KG .


Exercise 10.4Prove that the regular representation ρreg of G defined in Exampale 4(b)(2) corresponds to theregular KG-module KG˝ via Proposition 10.3.

Convention: In the sequel, when no confusion is to be made, we drop the ˝-notation to denote theregular KG-module and simply write KG instead of KG˝.

Lemma 10.5Two representations ρ1 : G ÝÑ GLpV1q and ρ2 : G ÝÑ GLpV2q are equivalent if and only if V1 – V2as KG-modules.

Proof : If ρ1 „ ρ2 and α : V1 ÝÑ V2 is a K -isomorphism such that ρ2pgq “ α ˝ ρ1pgq ˝ α´1 for each g P G,then by Proposition 10.3 for every v P V1 and every g P G we have

g ¨ αpvq “ ρ2pgqpαpvqq “ αpρ1pgqpvqq “ αpg ¨ vq ,

hence α is a KG-isomorphism. Conversely, if α : V1 ÝÑ V2 is a KG-isomorphism, then certainly it is aK -homomorphism and for each g P G and by Proposition 10.3 for each v P V2 we have

α ˝ ρ1pgq ˝ α´1pvq “ αpρ1pgqpα´1pvqq “ αpg ¨ α´1pvqq “ g ¨ αpα´1pvqq “ g ¨ v “ ρ2pgqpvq ,

hence ρ2pgq “ α ˝ ρ1pgq ˝ α´1 for each g P G.

Remark 10.6 (Dictionary)More generally, through Proposition 10.3, we may transport terminology and properties from KG-modules to K -representations and conversely.This lets us build the following translation dictionary:

K -Representations KG-Modules

K -representation of G ÐÑ KG-moduledegree ÐÑ K -rankhomomorphism of K -representations ÐÑ homomorphism of KG-modulesequivalent K -representations ÐÑ isomorphism of KG-modulessubrepresentation ÐÑ KG-submoduledirect sum of representations ρV1

‘ ρV2ÐÑ direct sum of KG-modules V1 ‘ V2

irreducible representation ÐÑ simple (“ irreducible) KG-modulethe trivial representation ÐÑ the trivial KG-module Kthe regular representation of G ÐÑ the regular KG-module KGcompletely reducible K -representation ÐÑ semisimple KG-module

(= completely reducible)every K -representation of G is ÐÑ KG is semisimplecompletely reducible. . . . . .


Finally we introduce an ideal of KG which encodes a lot of information about KG-modules.

Proposition-Definition 10.7 (The augmentation ideal)The map ε : KG ÝÑ K,

ř

gPG λgg ÞÑř

gPG λg is an algebra homomorphism, called augmentationhomomorphism (or map). Its kernel kerpεq “: IpKGq is an ideal and it is called the augmentationideal of KG. The following statements hold:

(a) IpKGq “ tř

gPG λgg P KG |ř

gPG λg “ 0u “ annKGpK q and if K is a field IpKGq Ě JpKGq ;

(b) KG{IpKGq – K as K -algebras;

(c) IpKGq is a free K -module of rank |G|-1 with K -basis tg´ 1 | g P Gzt1uu;

Proof : Clearly, the map ε : KG ÝÑ K is the unique extension by K -linearity of the trivial representationG ÝÑ Kˆ Ď K, g ÞÑ 1K to KG, hence is an algebra homomorphism and its kernel is an ideal of thealgebra KG.

(a) IpKGq “ kerpεq “ tř

gPG λgg P KG |ř

gPG λg “ 0u by definition of ε. The second equality isobvious by definition of annKGpK q, and the last inclusion follows from the definition of the Jacobsonradical.

(b) follows from the 1st isomorphism theorem.(c) Let

ř

gPG λgg P IpKGq. Thenř

gPG λg “ 0 and henceÿ

gPGλgg “

ÿ

gPGλgg´ 0 “

ÿ

gPGλgg´

ÿ

gPGλg “

ÿ

gPGλgpg´ 1q “

ÿ

gPGzt1uλgpg´ 1q ,

which proves that the set tg ´ 1 | g P Gzt1uu generates IpKGq as a K -module. The abovecomputations also show that

ÿ

gPGzt1uλgpg´ 1q “ 0 ùñ

ÿ

gPGλgg “ 0

Hence λg “ 0 @g P G, which proves that the set tg´1 | g P Gzt1uu is also K -linearly independent,hence a K -basis of IpKGq.

Lemma 10.8If K is a field of positive characteristic p and G is p-group, then IpKGq “ JpKGq.

Exercise 10.9 (Proof of Lemma 10.8. Proceed as indicated.)

(a) Recall that an ideal I of a ring R is called a nil ideal if each element of I is nilpotent. Acceptthe following result: if I is a nil left ideal in a left Artinian ring R then I is nilpotent.

(b) Prove that g ´ 1 is a nilpotent element for each g P Gzt1u and deduce that IpKGq is a nilideal of KG.

(c) Deduce from (a) and (b) that IpKGq Ď JpKGq using Exercise 3 on Exercise Sheet 2.

(d) Conclude that IpKGq “ JpKGq using Proposition-Definition 10.7.


11 Semisimplicity and Maschke’s Theorem

Throughout this section, we assume that K is a field.

Our first aim is to prove that the semisimplicity of the group algebra depends on both the characteristicof the field and the order of the group.

Theorem 11.1 (Maschke)If charpK q - |G|, then KG is a semisimple K -algebra.

Proof : By Theorem-Definition 6.2, we need to prove that every s.e.s. 0 L M N 0φ ψ of KG-modules splits. However, the field K is clearly semisimple (again by Proposition-Definition 6.2). Henceany such sequence regarded as a s.e.s. of K -vector spaces and K -linear maps splits. So let σ : N ÝÑ Mbe a K -linear section for ψ and set

rσ :“ 1|G|

ř

gPG g´1σg : N ÝÑ Mn ÞÑ 1

|G|ř

gPG g´1σpgnq.

We may divide by |G|, since charpK q - |G| implies that |G| P Kˆ. Now, if h P G and n P N , then

rσphnq “ 1|G|

ÿ

gPGg´1σpghnq “ h 1

|G|ÿ

gPGpghq´1σpghnq “ hrσpnq

andψrσpnq “ 1

|G|ÿ

gPGψ`

g´1σpgnq˘ ψ KG-lin

“1|G|

ÿ

gPGg´1ψσpgnq “ 1

|G|ÿ

gPGg´1gn “ n ,

where the last-but-one equality holds because ψσ “ IdN . Thus rσ is a KG-linear section for ψ.

Example 6If K “ C is the field of complex numbers, then CG is a semisimple C-algebra, since charpCq “ 0.

It turns out that the converse to Maschke’s theorem also holds, and follows from the properties of theaugmentation ideal.

Theorem 11.2 (Converse of Maschke’s Theorem)If KG is a semisimple K -algebra, then charpK q - |G|.

Proof : Set charpK q “: p and let us assume that p | |G|. In particular p must be a prime number. We haveto prove that then KG is not semisimple.Claim: If 0 ‰ V Ă KG is a KG-submodule of KG˝, then V X IpKGq ‰ 0.Indeed: Let v “

ř

gPG λgg P V zt0u. If εpvq “ 0 we are done. Else, set t :“ř

hPG h. Then

εptq “ÿ

hPG1 “ |G| “ 0

as charpK q | |G|. Hence t P IpKGq. Now consider the element tv . On the one hand tv P V since V is asubmodule of KG˝, and on the other hand tv P IpKGqzt0u since

tv“´

ÿ

hPGh¯´

ÿ

gPGλgg

¯

“ÿ

h,gPGp1K ¨λgqhg“

ÿ

xPG

´

ÿ

gPGλg¯

x“ÿ

xPGεpvqx ñ εptvq“

ÿ

xPGεpvq “ |G|εpvq“0 .


The Claim implies that IpKGq, which is a KG-submodule by definition, cannot have a complement in KG˝.Therefore, by Proposition 6.1, KG˝ is not semisimple and hence KG is not semisimple by Theorem-Definition 6.2.

In the case in which the field K is algebraically closed, or a splitting field for KG, the following exerciseoffers a second proof of the converse of Maschke’s Theorem exploiting the Artin-Wedderburn Theorem(Theorem 8.2).

Exercise 11.3 (Proof of the Converse of Maschke’s Theorem for K splitting field for KG.)Assume K is a field of positive characteristic p with p | |G| and is a splitting field for KG. SetT :“ x

ř

gPG gyK .

(a) Prove that we have a series of KG-submodules given by KG˝ Ľ IpKGq Ě T Ľ 0.

(b) Deduce that KG˝ has at least two composition factors isomorphic to the trivial module K .

(c) Deduce that KG is not a semisimple K -algebra using Theorem 8.2.

12 Simple modules over splitting fields

Assumption 12.1Throughout this section, we assume that K is a splitting field for KG, and wesimply say that K is a splitting field for G.

As explained at the end of Chapter 2 this assumption, slightly weaker thanrequiring that K “ K , implies that the conclusions of Theorem 8.2, Corollary 8.3and Corollary 8.4 still hold.

We state here some elementary facts about simple KG-modules, which we obtain as consequences ofthe Artin-Wedderburn structure theorem.

Corollary 12.2If K is a splitting field for G, then there are only finitely many isomorphism classes of simpleKG-modules.

Proof : The claim follows directly from Assumption 12.1 and Corollary 8.3.

Corollary 12.3If G is an abelian group and K is a splitting field for G, then any simple KG-module is one-dimensional.

Proof : Since KG is commutative the claim follows directly from Assumption 12.1 and Corollary 8.4.


Corollary 12.4Let p be a prime number. If G is a p-group, K is a splitting field for G and charpK q “ p, then thetrivial module is the unique simple KG-module, up to isomorphism.

Proof : By Lemma 10.8 we have JpKGq “ IpKGq. Thus KG{JpKGq – K as K -algebras by Proposition-Definition 10.7(b). Now, as K is commutative, Z pK q “ K , and it follows from Assumption 12.1 andCorollary 8.3 that

|MpKGq| “ dimK Z pKG{JpKGqq “ dimK K “ 1 .

Remark 12.5Another standard proof for Corollary 12.4 consists in using a result of Brauer’s stating that |MpKGq|equals the number of conjugacy classes of G of order not divisible by the characteristic of the field K .

Corollary 12.6If K is a splitting field for G and charpK q - |G|, then |G| “

ř

SPMpKGq dimK pSq2.

Proof : Since charpK q - |G|, the group algebra KG is semisimple by Maschke’s Theorem. Thus it followsfrom Assumption 12.1 and Theorem 8.2 that

ÿ

SPMpKGqdimK pSq2 “ dimK pKGq “ |G| .

Chapter 4. Operations on Groups and Modules

In this chapter we show how to construct new KG-modules from old ones using standard module op-erations such as tensor products, Hom-functors, duality, or using subgroups or quotients of the initialgroup. Moreover, we study how these constructions relate to each other.

Notation: throughout this chapter, unless otherwise specified, we let G denote a finite group and K bea commutative ring. All modules over group algebras considered are assumed to be free of finite rankas K -modules.

References:






13 Tensors, Hom’s and dualityDefinition 13.1 (Tensor product of KG-modules)

If M and N are two KG-modules, then the tensor product M bK N of M and N balanced over Kbecomes a KG-module via the diagonal action of G. In other words, the external composition lawis defined by the G-action

¨ : G ˆ pM bK Nq ÝÑ M bK Npg,mb nq ÞÑ g ¨ pmb nq :“ gmb gn


36


Definition 13.2 (Homs)If M and N are two KG-modules, then the abelian group HomK pM,Nq becomes a KG-module viathe so-called conjugation action of G. In other words, the external composition law is defined bythe G-action

¨ : G ˆHomK pM,Nq ÝÑ HomK pM,Nqpg, fq ÞÑ g ¨ f : M ÝÑ N,m ÞÑ pg ¨ fqpmq :“ g ¨ fpg´1 ¨mq


Specifying Definition 13.2 to N “ K yields a KG-module structure on the K -dual M˚ “ HomK pM,K q.

Definition 13.3 (Dual of a KG-module)

(a) If M is a KG-module, then its K -dual M˚ becomes a KG-module via the external compositionlaw is defined by the map

¨ : G ˆM˚ ÝÑ M˚

pg, fq ÞÑ g ¨ f : M ÝÑ K,m ÞÑ pg ¨ fqpmq :“ fpg´1 ¨mq


(b) If M,N are KG-modules, then every KG-homomorphism ρ P HomKGpM,Nq induces a KG-homomorphism

ρ˚ : N˚ ÝÑ M˚

f ÞÑ ρ˚pfq : M ÝÑ K,m ÞÑ ρ˚pfqpmq :“ f ˝ ρpmq .

(See Propotion D.3.)

Lemma 13.4If M and N are KG-modules, then HomK pM,Nq – M˚ bK N as KG-modules.

Proof : The map

θ :“ θM,N : M˚ bK N ÝÑ HomK pM,Nqf b n ÞÑ θpf b nq : M ÝÑ N,m ÞÑ θpf b nqpmq “ fpmqn

defines a K -isomorphism. (Check it!)Now, for every g P G, f P M˚, n P N and m P M , we have on the one hand

θpg ¨ pf b nqqpmq “ θpg ¨ f b g ¨ nqqpmq “ pg ¨ fqpmqg ¨ n“ fpg´1 ¨mqg ¨ n

and on the other hand`

g ¨ θpf b nq˘

pmq “ g ¨`

θpf b nqpg´1mq˘

“ g ¨`

fpg´1mqn˘

“ fpg´1 ¨mqg ¨ n ,

hence θpg ¨ pf b nqq “`

g ¨ θpf b nq˘

and it follows that θ is in fact a KG-isomorphism.


Remark 13.5In case M “ N the above constructions yield a KG-module structure on EndK pMq – M˚ bK M .Moreover, if rkK pMq “: n, tm1, . . . , mnu is a K -basis of M and tm˚1 , . . . , m˚nu is the dual K -basis,then IdM P EndK pMq corresponds to the element r :“

řni“1m˚i bmi P M˚ bK M . (Exercise!)

This allows us to define the KG-homomorphism:

I : K ÝÑ M˚ bK M1 ÞÑ r

Definition 13.6 (Trace map)If M is a KG-module, then the trace map associated to M is the KG-homomorphism

TrM : M˚ bK M ÝÑ Kf bm ÞÑ fpmq .

Notation 13.7If M and N are KG-modules, we shall write M | N to mean that M is isomorphic to a directsummand of N .

Lemma 13.8If rkK pMq P Kˆ, then K | M˚ bK M .

Proof : By Lemma-Definition D.4(c) it suffices to check that 1rkK pMq I is a KG-section for TrM , because then

M˚ bK M – kerpTrMq ‘ K , hence K | M˚ bK M . So let λ P K . Then

”

TrM ˝1

rkK pMqIı

pλq “ 1rkK pMq

TrMpλrq “λ

rkK pMqTrMp

nÿ

i“1m˚i bmiq

“λ

rkK pMq

nÿ

i“1m˚i pmiq

“λ

rkK pMq

nÿ

i“11 “ λ .

Hence TrM ˝ 1rkK pMq I “ IdK .

Exercise 13.9Assume K is a field (to simplify) and let M ,N be KG-modules. Prove the following assertions:

(a) M – pM˚q˚ as KG-modules (in a natural way);

(b) M˚ ‘N˚ – pM ‘Nq˚ and M˚ bK N˚ – pM bK Nq˚ as KG-modules (in a natural way);

(c) M is simple, resp. indecomposable, resp. semisimple, if and only if M˚ is simple, resp.indecomposable, resp. semisimple;

(d) TrM is a KG-homomorohism and TrM ˝ θ´1M,M coincides with the ordinary trace of matrices;

(e) M | MbK M˚bK M , and if char(K) | dimK pMq, then M‘M | MbK M˚bK M . (This is morechallenging!)


Can (a) to (e) be generalised to the case in which K is an arbitrary commutative ring and M andN free of finite rank when seen as K -modules ?

14 Fixed and cofixed pointsFixed and cofixed points explain why in the previous section we considered tensor products and Hom’sover K and not over KG.

Definition 14.1 (G-fixed points and G-cofixed points)Let M be a KG-module.

(a) The G-fixed points of M are by definition MG :“

m P M | g ¨m “ m @g P G(

.

(b) The G-cofixed points of M are by definition MG :“ M{pIpKGq ¨Mq.

In other words:

¨ MG is the largest KG-submodule of M on which G acts trivially, and

¨ MG is the largest quotient of M on which G acts trivially.

Lemma 14.2If M,N are KG-modules, then HomK pM,NqG “ HomKGpM,Nq and pM bKNqG – M bKG N .

Proof : A K -linear map f : M ÝÑ N is a morphism of KG-modules if and only if fpg ¨mq “ g ¨ fpmq for allg P G and all m P M , that is if and only if g´1 ¨ fpg ¨ mq “ fpmq for all g P G and all m P M . This isexactly the condition that f is fixed under the action of G. Hence HomK pM,NqG “ HomKGpM,Nq.Second claim: similar, Exercise!

Exercise 14.3

Let K be a field and let 0 ÝÑ L φÝÑ M ψ

ÝÑ N ÝÑ 0 be a s.e.s. of KG-modules. Prove that ifM – L‘N , then the s.e.s. splits.[Hint: Consider the exact sequence induced by the functor HomKGpN,´q (as in Proposition D.3(a)) and use the fact thatthe modules considered are all finite-dimensional.]

15 Inflation, restriction and inductionIn this section we define new module structures from known ones for subgroups, overgroups and quo-tients, and investigate how these relate to each other.

Remark 15.1(a) If H ď G is a subgroup, then the inclusion H ÝÑ G, h ÞÑ h can be extended by K -linearity

to an injective algebra homomorphism ı : KH ÝÑ KG,ř

hPH λhh ÞÑř

hPH λhh. Hence KH isa K -subalgebra of KG.


(b) Similarly, if U E G is a normal subgroup, then the quotient homomorphism G ÝÑ G{U ,g ÞÑ gU can be extended by K -linearity to an algebra homomorphism π : KG ÝÑ K rG{Us.

It is clear that we can always perform changes of the base ring using the above homomorphism in orderto obtain new module structures. This yields two natural operations on modules over group algebrascalled inflation and restriction.

Definition 15.2 (Restriction)Let H ď G be a subgroup. If M is a KG-module, then M may be regarded as a KH-module througha change of the base ring along ı : KH ÝÑ KG, which we denote by ResGHpMq or simply M ÓGH andcall the restriction of M from G to H .

Definition 15.3 (Inflation)Let U E G be a normal subgroup. If M is a K rG{Us-module, then M may be regarded as aKG-module through a change of the base ring along π : KG ÝÑ K rG{Us, which we denote byInfGG{UpMq and call the inflation of M from G{U to G.

Remark 15.4

(a) If H ď G is a subgroup, M is a KG-module and ρ : G ÝÑ GLpMq is the associatedK -representation, then the K -representation associated to M ÓGH is simply the compositemorphism

H ıÝÑ G ρ

ÝÑ GLpMq .

(b) Similarly, if U E G is a normal subgroup, M is a K rG{Us-module and ρ : G{U ÝÑ GLpMqis the associated K -representation, then the K -representation associated to InfGG{UpMq issimply

G πÝÑ G{U ρ

ÝÑ GLpMq .

Lemma 15.5

(a) If H ď G and M1,M2 are two KG-modules, then pM1 ‘M2q ÓGH “ M1Ó

GH ‘M2Ó

GH . If U E G

and M1,M2 are two K rG{Us-modules, then InfGG{UpM1 ‘M2q “ InfGG{UpM1q ‘ InfGG{UpM2q.

(b) (Transitivity of restriction) If L ď H ď G and M is a KG-module, then M ÓGHÓHL “ M ÓGL .

(c) If H ď G and M is a KG-module, then pM˚q ÓGH– pM ÓGHq˚. If U E G and M is a K rG{Us-module, then InfGG{UpM˚q – pInfGG{U Mq˚.

Proof :

(a) Straightforward from the fact that the external composition law on a direct sum is defined compo-nentwise.

(b) If ıL,H : L ÝÑ H denotes the canonical inclusion of L in H , ıH,G : H ÝÑ G the canonical inclusionof H in G and ıL,G : L ÝÑ G the canonical inclusion of L in G, then

ıH,G ˝ ıL,H “ ıL,G .


Thus performing a change of the base ring via ıL,G is the same as performing two successive changesof the base ring via first ıH,G and then ıL,H . Hence M ÓGHÓHL “ M ÓGL .

(c) Straightforward.

A third natural operation comes from extending scalars from a subgroup to the initial group.

Definition 15.6 (Induction)Let H ď G be a subgroup and let M be a KH-module. Regarding KG as a pKG,KHq-bimodule,we define the induction of M from H to G to be the left KG-module

IndGHpMq :“ KG bKH M .

We shall also write M ÒGH instead of IndGHpMq.

Example 7

(a) If H “ t1u and M “ K , then K ÒGt1u“ KG bK K – KG.

(b) Transitivity of induction: clearly L ď H ď G and M is a KL-module, then M ÒGL “ pM ÒHL qÒGHby the associativity of the tensor product.

First, we analyse the structure of an induced module in terms of the left cosets of H .

Remark 15.7Recall that G{H “ tgH | g P Gu denotes the set of left cosets of H in G. Moreover, we writerG{Hs for a set of representatives of these left cosets. In other words, rG{Hs “ tg1, . . . , g|G:H|u(where we assume that g1 “ 1) for elements g1, . . . , g|G:H| P G such that giH ‰ gjH if i ‰ j andG is the disjoint union of the left cosets of H , so that

G “ğ

gPrG{HsgH “ g1H \ . . .\ g|G:H|H .

It follows thatKG “

à

gPrG{HsgKH ,

where gKH “ tgř

hPH λhh | λh P K @h P Hu. Clearly, gKH – KH as right KH-modules viagh ÞÑ h for each h P H . Therefore

KG –à

gPrG{HsKH “ pKHq|G:H|

and hence is a free right KH-module with a KH-basis given by the left coset representativesin rG{Hs.

In consequence, if M is a given KH-module, then we have

KG bKH M “ pà

gPrG{HsgKHq bKH M “

à

gPrG{HspgKH bKH Mq “

à

gPrG{HspgbMq ,


where we setgbM :“ tgbm | m P Mu Ď KG bKH M .

Clearly, each gbM is isomorphic to M as a K -module via the K -isomorphism

gbM ÝÑ M,gbm ÞÑ m .

It follows thatrkK pIndGHpMqq “ |G : H| ¨ rkK pMq .

Next we see that with its left action on KG bKH M , the group G permutes these K -submodules:for if x P G, then xgi “ gjh for some h P H , and hence

x ¨ pgi bmq “ xgi bm “ gjhbm “ gj b hm .

This action is also clearly transitive since for every 1 ď i, j ď |G : H| we can write

gjg´1i pgi bMq “ gj bM .

Exercise: Check that the stabiliser of g1 bM is H (where g1 “ 1) and deduce that the stabiliserof gi bM is giHg´1

i .

Proposition 15.8 (Universal property of the induction)Let H ď G, let M be a KH-module and let j : M ÝÑ KG bKH M,m ÞÑ 1 b m be the canonicalmap (which is in fact a KH-homomorphism). Then, for every KG-module N and for every KH-homomorphism φ : M ÝÑ ResGHpNq, there exists a unique KG-homomorphism φ : KGbKHM ÝÑ Nsuch that φ ˝ j “ φ, or in other words such that the following diagram commutes:

M N

IndGHpMq

j

φ

ö

D! φ

Proof : The universal property of the tensor product yields the existence of a well-defined homomorphism ofabelian groups

φ : KG bKH M ÝÑ Nabm ÞÑ a ¨ φpmq .

which is obviously KG-linear. Moreover, for each m P M , we have φ˝jpmq “ φp1bmq “ 1¨φpmq “ φpmq,hence φ ˝ j “ φ. Finally the uniqueness follows from the fact for each a P KG and each m P M , we have

φpabmq “ φpa ¨ p1bmqq “ a ¨ φp1bmq “ a ¨ pφ ˝ jpmqq “ a ¨ φpmq

hence there is a unique possible definition for φ.

Induced modules can be hard to understand from first principles, so we now develop some formalismthat will enable us to compute with them more easily.


To begin with, there is, in fact, a further operation that relates the modules over a group G and asubgroup H called coinduction. Given a KH-module M , then the coinduction of M from H to G is theleft KG-module

CoindGHpMq :“ HomKHpKG,Mqwhere the left KG-module structure is defined through the natural right KG-module structure of KG:

¨ : KG ˆHomKHpKG,Mq ÝÑ HomKHpKG,Mqpg, θq ÞÑ g ¨ θ : KG ÝÑ M, x ÞÑ pg ¨ θqpxq :“ θpx ¨ gq

Example 8

If H “ t1u and M “ K , then CoindGt1upK q – pKGq˚ (i.e. with the KG-module structure of pKGq˚ ofDefinition 13.3).Exercise: exhibit a KG-isomorphism between the coinduction of K from t1u to G and pKGq˚.

Now, we see that the operation of coinduction in the context of group algebras is just a disguisedversion of the induction functor.

Lemma 15.9 (Induction and coinduction are the same)If H ď G is a subgroup and M is a KH-module, then KGbKHM – HomKHpKG,Mq as KG-modules.In particular, KG – pKGq˚ as KG-modules.

Proof : Mutually inverse KG-isomorphisms are defined by

Φ : KG bKH M ÝÑ HomKHpKG,Mqgbm ÞÑ Φpgbmq : KG ÝÑ M, x ÞÑ pxgqm

andΨ : HomKHpKG,Mq ÝÑ KG bKH M

θ ÞÑř

gPrG{Hs gb θpg´1q .

It follows that in the case in which H “ t1u and M “ K ,

KG – KG bK K – HomK pKG,K q – pKGq˚

as KG-modules. Here we emphasise that the last isomorphism isn’t an equality. See the previousexample.

Theorem 15.10 (Adjunction / Frobenius reciprocity / Nakayama relations)Let H ď G be a subgroup. Let N be a KG-module and let M be a KH-module. Then, there areK -isomorphisms:

(a) HomKHpM,HomKGpKG,Nqq – HomKGpKG bKH M,Nq,or in other words, HomKHpM,N ÓGHq – HomKGpM ÒGH , Nq ;

(b) HomKHpN ÓGH ,Mq – HomKGpN,M ÒGHq .

Proof :

(a) Since induction and coinduction coincide, we have HomKGpKG,Nq – KG bKG N – N as KG-modules. Therefore, HomKGpKG,Nq – N ÓGH as KH-modules, and it suffices to prove the secondisomorphism. In fact, this K -isomorphism is given by the map


Φ : HomKHpM,N ÓGHq ÝÑ HomKGpM ÒGH , Nqφ ÞÑ φ

where φ is the KG-homomorphism induced by φ by the universal property of the induction. Sinceφ is the unique KG-homomorphism such that φ ˝ j “ φ, setting

Ψ : HomKGpM ÒGH , Nq ÝÑ HomKHpM,N ÓGHqψ ÞÑ ψ ˝ j

provides us with an inverse map for Φ. Finally, it is straightforward to check that both Φ and Ψare K -linear.

(b) Exercise: check that the so-called exterior trace map

pTrGH : HomKHpN ÓGH ,Mq ÝÑ HomKGpN,M ÒGHq

φ ÞÑ pTrGHpφq : N ÝÑ M ÒGH , n ÞÑ

ř

gPrG{Hs gb φpg´1nq

provides us with the required K -isomorphism.

Proposition 15.11Let H ď G be a subgroup. Let N be a KG-module and let M be a KH-module. Then, there areKG-isomorphisms:

(a) pM bK N ÓGHqÒGH – M ÒGH bKN; and

(b) HomK pM,N ÓGHqÒGH – HomK pM ÒGH , Nq.

Proof : (a) It follows from the associativity of the tensor product that

pM bK N ÓGHqÒGH“ KG bKH pM bK N ÓGHq – pKG bKH Mq bK N “ M ÒGH bKN

(b) We push back the proof until we have introduced the concept of an H-free module. (We will thenprove that if M is a KH-module, then pM˚qÒGH– pM ÒGHq˚ and (b) will follow directly from (a) andthe KG-isomorphism of Lemma 13.4.)

Exercise 15.12Let K be a field. Let U,V ,W be KG-modules. Prove that there are isomorphisms of KG-modules:

(i) HomK pU bK V ,W q – HomK pU,V ˚ bK W q; and

(ii) HomKGpU bK V ,W q – HomKGpU,V ˚ bK W q – HomKGpU,HomK pV ,W qq.

Chapter 5. The Mackey Formula and Clifford Theory

The results in this chapter go more deeply into the theory. We start with the so-called Mackey de-composition formula, which provides us with yet another relationship between induction and restriction.After that we explain Clifford’s theorem, which explains what happens when a simple representation isrestricted to a normal subgroup. These results are essential and have many consequences throughoutrepresentation theory of finite groups.

Notation: throughout this chapter, unless otherwise specified, we let G denote a finite group and K bea commutative ring. All modules over group algebras considered are assumed to be free of finite rankas K -modules (hence, in particular, they are finitely generated as KG-modules).

References:






16 Double cosetsDefinition 16.1 (Double cosets)

Given subgroups H and L of G we define for each g P G

HgL :“ thgk P G | h P H, k P Lu

and call this subset of G the pH, Lq-double coset of g. Moreover, we let HzG{L denote the set ofpH, Lq-double cosets of G.

45


First, we want to prove that the pH, Lq-double cosets partition the group G.

Lemma 16.2Let H, L ď G.

(a) Each pH, Lq-double coset is a disjoint union of right cosets of H and a disjoint union of leftcosets of L.

(b) Any two pH, Lq-double cosets either coincide or are disjoint. Hence, letting rHzG{Ls denotea set of representatives for the pH, Lq-double cosets of G, we have

G “ğ

gPrHzG{LsHgL .

Proof :

(a) If hgk P HgL and k1 P L, then hgk ¨ k1 “ hgpkk1q P HgL. It follows that the entire left coset ofL that contains hgk is contained in HgL. This proves that HgL is a union of left cosets of L. Asimilar argument proves that HgL is a union of right cosets of H .

(b) Let g1, g2 P G. If h1g1k1 “ h2g2k2 P Hg1L X Hg2L, then g1 “ h´11 h2g2k2k´1

1 P Hg2L so thatHg1L Ď Hg2L. Similarly Hg2L Ď Hg1L. Thus if two double cosets are not disjoint, they coincide.

If X is a left G-set we use the standard notation GzX for the set of orbits of G on X , and denote a setof representatives for theses orbits by rGzX s. Similarly if Y is a right G-set we write Y {G and rY {Gs.We shall also repeatedly use the orbit-stabiliser theorem without further mention: in other words, if Xis a transitive left G-set and x P X then X – G{StabGpxq (i.e. the set of left cosets of the stabiliser ofx in G), and similarly for right G-sets.

Exercise 16.3

(a) Let H, L ď G. Prove that the set of pH, Lq-double cosets is in bijection with the set of orbitsHzpG{Lq, and also with the set of orbits pHzGq{L under the mappings

HgL ÞÑ HpgLq P HzpG{Lq

HgL ÞÑ pHgqL P pHzGq{L.

This justifies the notation HzG{L for the set of pH, Lq-double cosets.

(b) Let G “ S3. Consider H “ L :“ S2 “ tId, p1 2qu as a subgroup of S3. Prove that

rS2zS3{S2s “ tId, p1 2 3qu

whileS2zS3{S2 “ t tId, p1 2qu, tp1 2 3q, p1 3 2q, p1 3q, p2 3quu .


17 The Mackey formulaIf H and L are subgroups of G, we wish to describe what happens if we induce a KL-module from L toG and then restrict it to H .

Remark 17.1We need to examine KG seen as a pKH,KLq-bimodule (i.e. with left and right external laws bymultiplication in G). Since G “

Ů

gPrHzG{LsHgL, we have

KG “à

gPrHzG{LsK xHgLy

as pKH,KLq-bimodule, where K xHgLy denotes the free K -module with K -basis HgL.Now if M is a KL-module, we will also write gM for gbM , which is a left K p gLq-module with

pgkg´1q ¨ pgbmq “ gb km

for each k P L and each m P M . With this notation, we have

K xHgLy – KH bK pHX gLq pgb KLq ,

where hgk P HgL corresponds to hb gb k .

Theorem 17.2 (Mackey formula)Let H, L ď G and let M be a KL-module. Then, as KH-modules,

M ÒGLÓGH –à

gPrHzG{Lsp gM Ó gLHX gLqÒ

HHX gL .

Proof : It follows from Remark 17.1 that as left KH-modules we have

M ÒGLÓGH – pKG bKL MqÓGH –à

gPrHzG{LsK xHgLy bKL M

–à

gPrHzG{LsKH bK pHX gLq pgb KLq bKL M

–à

gPrHzG{LsKH bK pHX gLq pgbMqÓ

gLHX gL

–à

gPrHzG{Lsp gM Ó gL

HX gLqÒHHX gL .

Remark 17.3Given an arbitrary finite group Z , write KZmod for the category of KZ-modules which are free offinite rank as K -modules. Then, expressed in categorical terms, the Mackey formula says that wehave the following equality of functors from KLmod to KHmod:

ResGH ˝ IndGL “à

gPrHzG{LsIndHHX gL ˝Res gL

HX gL ˝Innpgq

where Innpgq is conjugation by g P G.


Exercise 17.4Let H, L ď G, let M be a KL-module and let N be a KH-module. Use the Mackey formula toprove that:

(a) M ÒGL bKN ÒGH–À

gPrHzG{LspgM Ó gLHX gL bKN ÓHHX gLqÒ

GHX gL ;

(b) HomK pM ÒGL , N ÒGHq –À

gPrHzG{LspHomK pgM Ó gLHX gL, N ÓHHX gLqqÒ

GHX gL .

18 Clifford theoryWe now turn to Clifford’s theorem, which we present in a weak and a strong form. Clifford theory is acollection of results about induction and restriction of simple modules from/to normal subgroups.

Throughout this section, we assume that K is a field.

First we emphasise again, that this is no loss of generality: indeed if S were a simple KG-modulewith K an arbitrary commutative ring, then letting I be the annihilator in K of S, we have that I is amaximal ideal of K , so that K {I is a field and S is a pK {IqG-module.

Theorem 18.1 (Clifford’s Theorem, weak form)If U E G is a normal subgroup and S is a simple KG-module, then S ÓGU is semisimple.

Proof : Let V be any simple KU-submodule of S ÓGU . Now, notice that for every g P G, gV :“ tgv | v P V uis also a KU-submodule of S ÓGU , since U E G for any u P U , we have

u ¨ gV “ g ¨ pg´1ugqlooomooon

PU

V “ gV

Moreover, gV is also simple, since if W were a non-trivial proper KU-submodule of gV then g´1Wwould also be a non-trivial proper submodule of g´1gV “ V . Now

ř

gPG gV is non-zero and it is aKG-submodule of S, which is simple, hence

ř

gPG gV “ S. Restricting to U , we obtain that

S ÓGU“ÿ

gPGgV

is a sum of simple KU-submodules. Hence S ÓGU is semisimple.

Remark 18.2The KU-submodules gV which appear in the proof of Theorem 18.1 are isomorphic to modules wehave seen before: more precisely the map

gb V ÝÑ gVgb v ÞÑ gv

is a KU-isomorphism, since U E G implies that gU “ U and hence the action of U on gb V (seeRemark 17.1) and gV is prescribed in the same way.


Theorem 18.3 (Clifford’s Theorem, strong form)Let U E G be a normal subgroup and let S be a simple KG-module. Then we may write

S ÓGU“ Sa11 ‘ ¨ ¨ ¨ ‘ Sarr

where r P Zą0 and S1, . . . , Sr are pairwise non-isomorphic simple KU-modules, occurring withmultiplicities a1, . . . , ar respectively. Moreover, the following statements hold:

(i) the group G permutes the homogeneous components of S ÓGU transitively;

(ii) a1 “ a2 “ ¨ ¨ ¨ “ ar and dimK pS1q “ ¨ ¨ ¨ “ dimK pSrq; and

(iii) S – pSa11 qÒ

GH1

as KG-modules, where H1 “ StabGpSa11 q.

Proof : The fact that S ÓGU is semisimple and hence can be written as a direct sum as claimed follows fromTheorem 18.1. Moreover, by the chapter on semisimplicity of rings and modules, we know that for each1 ď i ď r the homogeneous component Saii is characterised by Proposition 7.1: it is the unique largestKU-submodule which is isomorphic to a direct sum of copies of Si.

Now, if g P G then gpSaii q “ pgSiqai , where gSi is a simple KU-submodule of S ÓGU (see the proofof the weak form of Clifford’s Theorem). Hence there exists an index 1 ď j ď r such that gSi “ Sj andgpSaii q Ď Sajj (alternatively to Proposition 7.1, the theorem of Krull-Schmidt can also be invoked here).Because dimK pSiq “ dimK pgSiq, we have that ai ď aj . Similarly, since Sj “ g´1Si, we obtain aj ď aj .Hence ai “ aj holds. Because

S ÓGU“ gpS ÓGUq “ gpSa11 q ‘ ¨ ¨ ¨ ‘ gpS

arr q ,

we actually have that G permutes the homogeneous components. Moreover, asř

gPG gpSa11 q is a non-

zero KG-submodule of S, which is simple, we have thatř

gPG gpSa11 q “ S, and so the action on the

homogeneous components is transitive. This establishes both (i) and (ii).For (iii), we define a K -homomorphism via the map

Φ : pSa11 qÒ

GH1“ KG bKH1 S

a11 “

À

gPrG{H1sgb Sa1

1 ÝÑ Sgbm ÞÑ gm

that is, where gbm P gbSa11 . This is in fact a KG-homomorphism. Furthermore, the K -subspaces gpSa1

1 qof S are in bijection with the cosets G{H1, since G permutes them transitively by (i), and the stabiliserof one of them is H1. Thus both KGbKH1 S

a11 and S are the direct sum of |G : H1| K -subspaces gbSa1

1and gpSa1

1 q respectively, each K -isomorphic to Sa11 (via gbmØ m and gmØ m). Thus the restriction

of Φ to each summand is an isomorphism, and so Φ itself must be bijective, hence a KG-isomorphism.

One application of Clifford’s theory is for example the following Corollary:

Corollary 18.4Assume K is a splitting field for G of arbitrary characteristic. If p is a prime number and G is ap-group, then every simple KG-module has the form X ÒGH , where X is a 1-dimensional KH-modulefor some subgroup H ď G.

Proof : We proceed by induction on |G|.If |G| “ 1 or |G| is a prime number, then G is abelian and any simple module S is 1-dimensional byCorollary 12.3, so H “ G, X “ S and we are done.


So assume |G| “ pb with b ą 1, and let S be a simple KG-module and consider the subgroup

U :“ tg P G | g ¨ x “ x @ x P Su .

This is obviously a normal subgroup of G since it is the kernel of the K -representation associated to S.Hence S “ InfGG{UpT q for a simple K rG{Us-module T .Now, if U ‰ t1u, then |G{U| ă |G|, so by the induction hypothesis there exists a subgroup H{U ď G{Uand a 1-dimensional K rH{Us-module Y such that T “ IndG{UH{UpY q. But then

S “ InfGG{UpT q “ InfGG{U ˝ IndG{UH{UpY q “ IndGH ˝ InfHH{UpY q ,

so that setting X :“ InfHH{UpY q yields the result. Thus we may assume U “ t1u.If G is abelian, then all simple modules are 1-dimensional, so we are done. Assume now that G is notabelian. Then G has a normal abelian subgroup A that is not central. Indeed, to construct this subgroupA, let Z2pGq denote the second centre of G, that is, the preimage in G of Z pG{Z pGqq (this centre isnon-trivial as G{Z pGq is a non-trivial p-group). If x P Z2pGqzZ pGq, then A :“ xZ pGq, xy is a normalabelian subgroup not contained in Z pGq. Now, applying Clifford’s Theorem yields:

S ÓGA“ Sa11 ‘ ¨ ¨ ¨ ‘ Sarr

where r P Zą0, S1, . . . , Sr are non-isomorphic simple KA-modules and S “ pSa11 q Ò

GH1

, where H1 “StabGpSa1

1 q. We argue that V :“ Sa11 must be a simple KH1-module, since if it had a proper non-

trivial submodule W , then W ÒGH1would be a proper non-trivial submodule of S, which is simple: a

contradiction. If H1 ‰ G then by the induction hypothesis V “ X ÒH1H , where H ď H1 and X is a

1-dimensional KH-module. Thefore, by transitivity of the induction, we have

S “ pSa11 qÒ

GH1“ pX ÒH1

H qÒGH1“ X ÒGH ,

as required.Finally, the case H1 “ G cannot happen. For if it were to happen then

S ÓGA“ S ÓH1A “ Sa1

1 ,

is simple by the weak form of Clifford’s Theorem, hence of dimension 1 since A is abelian. The elementsof A must therefore act via scalar multiplication on S. Since such an action would commute with theaction of G, which is faithful on S, we deduce that A Ď Z pGq, which contradicts the construction of A.

Remark 18.5This result is extremely useful, for example, to construct the complex character table of a p-group.Indeed, it says that we need look no further than induced linear characters. In general, a KG-module of the form N ÒGH for some subgroup H ď G and some 1-dimensional KH-module is calledmonomial. A group all of whose simple CG-modules are monomial is called an M-group. (By theabove p-groups are M-groups.)

Chapter 6. Projective Modules over the Group Algebra

We continue developing techniques to describe modules that are not semisimple and in particular inde-composable modules. The indecomposable projective modules are the indecomposable summands of theregular module. Since every module is a homomorphic image of a direct sum of copies of the regularmodule, by knowing the structure of the projectives we gain some insight into the structure of all modules.

Notation: throughout this chapter, unless otherwise specified, we let G denote a finite group and Kbe a field. (We will understand in the coming chapters, why it is enough for our purpose to focus onfields, when considering projective module over the group algebra.) All modules over group algebrasconsidered are assumed to be finite-dimensional over K , hence finitely generated as KG-modules.When no confusion is to be made, we denote the regular module simply by KG instead of KG˝.

References:






19 Radical, socle, headBefore focusing on projective modules, at this point we examine further the structure of KG-moduleswhich are not semisimple, and try to establish connections with their semisimple submodules, semisimplequotients, and composition factors. This leads us to the definitions of the radical and the socle of amodule.

51


Definition 19.1Let M be a KG-module.

(a) The radical of M is its submodule radpMq :“Ş

VPMaxpMq V where MaxpMq denotes the setof maximal KG-submodules of M .

(b) The head of M is the quotient module hdpMq :“ M{ radpMq.

(c) The socle of M , denoted socpMq is the sum of all simple submodules of M .

Notice that in our setting (i.e. K is a field) KG-modules are always Noetherian, so provided M ‰ 0 theabove intersection is non-empty and hence radpMq ‰ M . Also, informally (talks/spoken mathematics)one also uses the words top and bottom instead of head and socle, respectively.

Lemma 19.2Let M be a KG-module. Then the following KG-submodules of M are equal:

(1) radpMq;

(2) JpKGqM;

(3) the smallest KG-submodule of M with semisimple quotient.

Proof :

“(3)“(1)”: Recall that if V Ă M is a maximal submodule, then M{V is simple. Moreover, if V1, . . . , Vr (r P Zą0)are maximal submodules of M , then the map

φ : M ÝÑ M{V1 ‘ ¨ ¨ ¨ ‘M{Vrm ÞÑ pm` V1, . . . , m` Vrq

is a KG-homomorphism with kernel kerpφq “ V1 X ¨ ¨ ¨ X Vr . Hence M{pV1 X ¨ ¨ ¨ X Vrq – Impφq issemisimple, since it is a submodule of a semisimple module. Therefore M{ radpMq is a semisimplequotient. It remains to see that it is the smallest such quotient.If X Ď M is a KG-submodule with M{X semisimple, then by the Correspondence Theorem, thereexists KG-submodules X1, . . . , Xr of M (r P Zą0) containing X such that

M{X – X1{X ‘ ¨ ¨ ¨ ‘ Xr{X and Xi{X is simple @ 1 ď i ď r .

For each 1 ď i ď r, let Yi be the kernel of the projection homomorphism M � M{X � Xi{X , sothat Yi is maximal (as Xi{X is simple) and X “ Y1 X . . .X Yr . Thus X Ě radpMq, as required.

“(1)Ď(2)”: Observe that the quotient module M{JpKGqM is a KG{JpKGq-module as

JpKGq pM{JpKGqMq “ 0 .

Now, as KG{JpKGq is semisimple (by Proposition 6.6 and Proposition 6.7), M{JpKGqM is a semisim-ple KG{JpKGq-module by definition of a semisimple ring, but then it is also semisimple as a KG-module. Since we have already proved that radpMq is the smallest KG-submodule of M withsemisimple quotient, we must have that radpMq Ď JpKGqM .

“(2)Ď(1)”: If Z Ď M is any KG-submodule for which M{Z is semisimple, certainly JpKGq ¨M{Z “ 0, becauseJpKGq annihilates all simple KG-modules by definition, and it follows that JpKGqM Ď Z . Thus,again as we already know that (1)“(3), we obtain that JpKGqM Ď radpMq.


Example 9If M is a semisimple KG-module, then socpMq “ M by definition, radpMq “ 0 by the above Lemma,and hence hdpMq “ M .

Lemma 19.3Let M be a KG-module. Prove that the following KG-submodules of M are equal:

(1) socpMq;

(2) the largest semisimple KG-submodule of M;

(3) tm P M | JpKGq ¨m “ 0u.

Proof : Exercise. [Hint: tm P M | JpKGq ¨m “ 0u is the largest KG-submodule of M annihilated by JpKGq,and hence may be seen as a KG{JpKGq-module.]

Remark 19.4 (Socle, radical and Loewy layers)We can iterate the notions of socle and radical: for each KG-module M and each n P Zě2 we defineinductively

radnpMq :“ rad´

radn´1pMq

¯

and socnpMq{ socn´1pMq :“ socpM{ socn´1pMqq

where we understand that rad1pMq “ radpMq and soc1pMq “ socpMq.

Exercise. Prove that:

(a) radnpMq “ JpKGqn ¨M and socnpMq “ tm P M | JpKGqn ¨m “ 0u ;

(b) ¨ ¨ ¨ Ď rad3pMq Ď rad2

pMq Ď radpMq Ď M and 0 Ď socpMq Ď soc2pMq Ď soc3pMq Ď ¨ ¨ ¨

The chains of submodules in (b) are called respectively, the radical series and socle series of M .The radical series of M is also known as the Loewy series of M . The quotients radn´1

pMq{ radnpMqare called the radical layers, or Loewy layers of M , and the quotients socnpMq{ socn´1pMq arecalled the socle layers of M .

Exercise 19.5Let M and N be KG-modules. Prove the following assertions.

(a) For every n P Zě1, radnpM‘Nq – radnpMq‘radnpNq and socnpM‘Nq – socnpMq‘socnpNq.

(b) The radical series of M is the fastest descending series of KG-submodules of M with semisim-ple quotients, and the socle series of M is the fastest ascending series of M with semisimplequotients. The two series terminate, and if r and n are the least integers for which radrpMq “ 0and socnpUq “ M then r “ n.

Definition 19.6The common length of the radical series and socle series of a KG-module M is called the Loewylength of M . (By the above, we may see it as the least integer n such that JpKGqn ¨M “ 0.)


20 Projective modulesFor the sake of clarity, we recall the general definition of a projective module through its most standardequivalent characterisations.

Proposition-Definition 20.1 (Projective module)Let R be an arbitrary ring and let P be an R-module. Then the following are equivalent:

(a) The functor HomRpP,´q is exact. In other words, the image of any s.e.s. of R-modules underHomRpP,´q is again a s.e.s.

(b) If ψ P HomRpM,Nq is a surjective morphism of R-modules, then the morphism of abeliangroups ψ˚ : HomRpP,Mq ÝÑ HomRpP,Nq is surjective. In other words, for every pair ofR-morphisms

P

M N

α

ψ

where ψ is surjective, there exists a KG-morphism β : P ÝÑ M such that α “ ψβ.

(c) If π : M ÝÑ P is a surjective R-homomorphism, then π splits, i.e., there exists σ P

HomRpP,Mq such that π ˝ σ “ IdP .

(d) The module P is isomorphic to a direct summand of a free R-module.

If P satisfies these equivalent conditions, then P is called projective. Moreover, a projective inde-composable module is called a PIM of R .

Example 10

(a) Any free module is projective.

(b) If e is an idempotent element of the ring R , then R – Re ‘ Rp1 ´ eq and Re is projective,but not free if e ‰ 0, 1.

(c) It follows from condition (d) of Proposition-Definition B.5 that a direct sum of modules tPiuiPIis projective if and only if each Pi is projective.

(d) If R is semisimple, then on the one hand any projective indecomposable module is simple, andconversely, since R˝ is semisimple. It follows that any R-module is projective.

21 Projective modules for the group algebraWe have seen that over a semisimple ring, all simple modules appear as direct summands of the regularmodule with multiplicity equal to their dimension. For non-semisimple rings this is not true any more,but replacing simple modules by the projective modules, we will obtain a similar characterisation.


To begin with we review a series of properties of projective KG-modules with respect to the operationson groups and modules we have introduced in Chapter 4, i.e. induction/restriction, tensor products, . . .

Proposition 21.1Assume K is an arbitrary commutative ring. Then the following assertions hold.

(a) If P is a projective KG-module and M is an arbitrary KG-module which is free of finite rankas a K -module, then P bK M is projective.

(b) If P is a projective KG-module and H ď G, then P ÓGH is a projective KH-module.

(c) If H ď G and P is a projective KH-module, then P ÒGH is a projective KG-module.

Proof :

(a) Since P is projective, by definition it is a direct summand of a free KG-module, so there exist aKG-module P 1 and a positive integer n such that P ‘ P 1 – pKGqn. Therefore,

pKGqn bK M – pP ‘ P 1q bK M – P bK M ‘ P 1 bK M

and it suffices to prove that pKGqnbKM is free. So observe that Example 7(a), Proposition 15.11(a)and the properties of the tensor product yield

KG bK M – pK ÒGt1uq bK M – pK bK M ÓGt1uqÒGt1u – M ÒGt1u– pK rkK pMqqÒGt1u– pK ÒGt1uqrkK pMq – pKGqrkK pMq

since M ÓGt1u is just M seen as K -module, and, as such, is free of finite rank. It follows immediately

that pKGqn bK M – pKGqn¨rkK pMq is a free KG-module, as required.(b) We have already seen that as a KH-module,

KG ÓGH – KH ‘ ¨ ¨ ¨ ‘ KH

where KH occurs with multiplicity |G : H|, so KG ÓGH is a free KH-module. Hence the restrictionfrom G to H of any free KG-module is a free KH-module. Now, by definition P | F for some freeKG-module F , so that P ÓGH | F ÓGH and the claim follows.

(c) Exercise![Hint: prove that KH ÒGH– KG.]

We now want to prove that the PIMs of KG are in bijection with the simple KG-modules, and hencethat there are a finite number of them, up to isomorphism. We will then be able to deduce from thisbijection that each of them occurs in the decomposition of the regular module with multiplicity equalto the dimension of the corresponding simple module.

Theorem 21.2

(a) If P is a projective indecomposable KG-module, then P{ radpPq is a simple KG-module.

(b) If M is a KG-module and M{ radpMq – P{ radpPq for a projective indecomposable KG-module P , then there exists a surjective KG-homomorphism φ : P ÝÑ M . In particular, if Mis also projective indecomposable, then M{ radpMq – P{ radpPq if and only if M – P .


(c) There is a bijection projective indecomposable

KG-modules(

{ –„ÐÑ

simpleKG-modules

(

{ –

P ÞÑ P{ radpPq

and hence the number of pairwise non-isomorphic PIMs of KG is finite.

Proof :

(a) By Lemma 19.2, P{ radpPq is semisimple, hence it suffices to prove that it is indecomposable, orequivalently, by Proposition 5.4 that EndKGpP{ radpPqq is a local ring.Now, if φ P EndKGpPq, then by Lemma 19.2, we have

φpradpPqq “ φpJpKGqPq “ JpKGqφpPq Ď JpKGqP “ radpPq .

Therefore, by the universal property of the quotient, φ induces a unique KG-homomorphismφ : P{ radpPq ÝÑ P{ radpPq such that the following diagram commutes:

P P

P{ radpPq P{ radpPq

πP

φ

πPö

φ

Then, the map

Φ : EndKGpPq ÝÑ EndKGpP{ radpPqqφ ÞÑ φ

is clearly a K -algebra homomorphism. Moreover Φ is surjectiv. Indeed, if ψ P EndKGpP{ radpPqq,then by the definition of a projective module there exists a KG-homomorphism φ : P ÝÑ P suchthat ψ ˝ πP “ πP ˝ φ. But then ψ satisfies the diagram of the universal property of the quotientand by uniqueness ψ “ φ.Finally, as P is indecomposable EndKGpPq is local, hence any element of EndKGpPq is eithernilpotent or invertible, and by surjectivity of Φ the same holds for EndKGpP{ radpPqq, which in turnmust be local.

(b) Consider the diagramP

M M{ radpMq P{ radpPq

πP

πM ψ–

where πM and πP are the quotient morphisms. As P is projective, by definition, there exists aKG-homomorphism φ : P ÝÑ M such that πP “ ψ ˝ πM ˝ φ.It follows that M “ φpPq ` radpMq “ φpPq ` JpKGqM , so that φpPq “ M by Nakayama’s Lemma.Finally, if M is a PIM, the surjective homomorphism φ splits by definition of a projective module,and hence M | P . But as both modules are indecomposable, we have M – P . Conversely, ifM – P , then clearly M{ radpMq – P{ radpPq.

(c) The given map between the two sets is well-defined by (a) and (b), and it is injective by (b). Itremains to prove that it is surjective. So let S be a simple KG-module. As S is finitely generated,there exists a free KG-module F and a surjective KG-homomorphism ψ : F ÝÑ S. But thenthere is an indecomposable direct summand P of F such that ψ|P : P ÝÑ S is non-zero, hence


surjective as S is simple. Clearly radpPq Ď kerpψ|Pq since it is the smallest KG-submodule withsemisimple quotient by Lemma 19.2. Then the universal property of the quotient yields a surjectivehomomorphism P{ radpPq ÝÑ S induced by ψ|P . Finally, as P{ radpPq is simple, P{ radpPq – Sby Schur’s Lemma.

Definition 21.3 (Projective cover of a simple module)If S is a simple KG-module, then we denote by PS the projective indecomposable KG-modulecorresponding to S through the bijection of Theorem 21.2(c) and call this module the projectivecover of S.

Corollary 21.4Assume K is a splitting field for G. Then, in the decomposition of the regular module KG intoa direct sum of indecomposable KG-submodules, each isomorphism type of projective indecompos-able KG-module occurs with multiplicity dimK pP{ radpPqq. In other words, with the notation ofDefinition 21.3,

KG –à

S simplepPSqns

where more preciely S runs through the set of isomorphism classes of simple KG-modules andnS “ dimK S.

Proof : Let KG “ P1 ‘ ¨ ¨ ¨ ‘ Pr (r P Zą0) be such a decomposition. In particular, P1, . . . Pr are PIMs. Then

JpKGq “ JpKGqKG “ JpKGqP1 ‘ ¨ ¨ ¨ ‘ JpKGqPr “ radpP1q ‘ ¨ ¨ ¨ ‘ radpPrq

by Lemma 19.2. Therefore,

KG{JpKGq – P1{ radpP1q ‘ ¨ ¨ ¨ ‘ Pr{ radpPrq

where each summand is simple by Theorem 21.2(a). Now as KG{JpKGq is semisimple, by Theorem 8.2,any simple KG{JpKGq-module occurs in this decomposition with multiplicity equal to its K -dimension.Thus the claim follows from the bijection of Theorem 21.2(c).

The Theorem also leads us to the following important dimensional restriction on projective modules.

Corollary 21.5Assume K is a splitting field for G of characteristic p ą 0. If P is a projective KG-module, then

|G|pˇ

ˇ dimK pPq .

(Here |G|p is the p-part of |G|, i.e. the exact power of p that divides the order of G.)

Proof : Let Q P SylppGq be a Sylow p-subgroup of G. By Lemma 21.1, P ÓGQ is projective. Moreover, byCorollary 12.4 the trivial KQ-module is the unique simple KQ-module, hence by Theorem 21.2(c) KQhas a unique PIM, namely KQ itself, which has dimension |Q| “ |G|p. Hence

P ÓGQ– pKQqm for some m P Zą0 .

Therefore,dimK pPq “ dimK pP ÓGQq “ m ¨ dimK KQ “ m ¨ |Q| “ m ¨ |G|p

and the claim follows.


22 The Cartan matrixNow that we have classified the projective KG-modules we turn to one of their important uses, whichis to determine the multiplicity of a simple module S as a composition factor of an arbitrary finitelygenerated KG-module M (hence with a composition series). We recall that if

0 “ M0 Ă M1 Ă M2 Ă ¨ ¨ ¨ Ă Mn´1 Ă Mn “ M

is any composition series of M , the number of quotients Mi{Mi´1 (1 ď i ď n) isomorphic to S isdetermined independently of the choice of composition series, by the Jordan–Hölder theorem. We callthis number the multiplicity of S in M as a composition factor.

Proposition 22.1Let S be a simple KG-module and let PS be its projective cover.

(a) If T is a simple KG-module then

dimK HomKGpPS , T q “#

dimK EndKGpSq if S – T ,0 if S fl T .

(b) If M is an arbitrary KG-module, then the multiplicity of S as a composition factor of M is

dimK HomKGpPS ,Mq{ dimK EndKGpSq .

Proof : Exercise, Sheet 4. [Hint: (b) can be proved by induction on the composition length of M .]

Definition 22.2

(a) If S and T are simple KG-modules, then the integer

cST :“ multiplicity of S as a composition factor of PT

is called the Cartan invariant associated to the pair pS, T q.

(b) The matrix C :“ pcST q with rows and columns indexed by the isomorphism classes of simpleKG-modules is called the Cartan matrix of KG.

It follows immediately from Proposition 22.1 that the Cartan invariants can be computed as follows.

Corollary 22.3If S and T are two simple KG-modules, then

cST “ dimK HomKGpPS , PT q{ dimK EndKGpSq .

In particular, if the ground field K is a splitting field for G, then

cST “ dimK HomKGpPS , PT q .


We will see later that there is an extremely effective way of computing the Cartan matrix using anothermatrix associated to the simple KG-modules, called the decomposition matrix.

23 Symmetry of the group algebraWe now want to obtain information about projective KG-modules using duality.Recall that we have already proved in Lemma 15.9 that pKGq˚ – KG as KG-modules. This allows usto deduce that projectivity is preserved by taking duals.

Proposition 23.1If P is a KG-module, then P is projective if and only if P˚ is.

Proof : As P˚˚ – P as KG-modules it suffices to prove one implication. Now, if P is a direct summandof pKGqn (n P Zě1), then P˚ is a direct summand of ppKGqnq˚ – ppKGq˚qn – pKGqn, and so is alsoprojective.

Next we want to investigate the relationship between head and socle of projective modules. For thispurpose, we recall the following properties of submodules, quotients and duality, seen in the Exerciseclasses:

W ď V KG-submodule ñ V ˚ has a KG-submodule M such that M – pV {W q˚ and V ˚{M – W ˚

Corollary 23.2Every projective indecomposable KG-module P has a simple socle. More precisely,

socpPq – pP˚{ radpP˚qq˚ .

Proof : As on the one hand P˚{ radpP˚q is simple by Theorem 21.2 and on the other hand radpP˚q is thesmallest KG-submodule of P˚ with semisimple quotient by Lemma 19.2, its dual is the largest semisimpleKG-submodule of P˚˚ – P , hence isomorphic to socpPq, which has to be simple, as required.

Alternatively, we could argue that as the socle is by definition the sum of all simple submodules, itsuffices to prove that P has a unique simple KG-submodule. Because P˚ is projective by Proposition 23.1,if S is any simple KG-module, then by duality the KG-homomorphisms S ÝÑ P are in bijection withthe KG-homomorphisms P˚ ÝÑ S˚ and it follows that

dimK HomKGpS,Pq “ dimK HomKGpP˚, S˚q

Moroever S˚ is also simple. Thus it follows from Proposition 22.1(a) that

dimK HomKGpP˚, S˚q “#

dimK EndKGpS˚q if P˚ is the projective cover of S˚,0 else.

Therefore, the claim follows from the fact that dimK EndKGpS˚q “ dimK EndKGpSq (again by duality).


Notice that in this proof to see that dimK HomKGpS,Pq “ dimK HomKGpP˚, S˚q and dimK EndKGpS˚q “dimK EndKGpSq it is also possible to argue using Lemma 13.4 and Lemma 14.2.

In fact, we can obtain a more precise statement and prove that the head and the socle of a PIM areisomorphic. For this purpose, we need the fact that the group algebra is a symmetric algebra.

Remark 23.3The map

p , q : G ˆ G ÝÑ K, pg, hq :“ δg,h´1

extended by K -bilinearity top , q : KG ˆ KG Ñ K

defines a K -bilinear form, which is symmetric, non-degenerate and associative. (Associative meansthat pab, cq “ pa, bcq @ a, b, c P KG).More generally, a K -algebra endowed with such a symmetric, non-degenerate and associativeK -bilinear form is called a symmetric algebra.

Theorem 23.4If P is a projective indecomposable KG-module, then P{ radpPq – socpPq.

Proof : Put S :“ P{ radpPq, which we know is simple as P is a PIM of KG, and assume S fl socpPq. WriteKG “ R ‘ Q, where Q “ Pn (n P Zą0) is the direct sum of all the indecomposable direct summands ofKG isomorphic to P and P - R . Then

socpQq – socpPqn

and Q does not contain any KG-submodule isomorphic to S. Next, consider the sum of all KG-submodulesof KG isomorphic to S and denote it by I , so that clearly 0 ‰ I Ă R and I is a left ideal of KG. However,as ψpIq Ď I for every ψ P EndKGpKGq, I is an ideal of KG. Now set J :“ tψ P EndKGpKGq | Impψq Ă Iu,so that J is clearly an ideal of EndKGpKGq. Let π : KG Ñ Q be the projection onto Q with kernel R .Then:

φ P J ñ radpKGq Ď kerpφqas the image of φ is semisimple, because it is a KG-submodule of I , which is itself a KG-submodule.Thus, φ|R “ 0, as S - hdpRq, and it follows that

φ ˝ π “ φ und π ˝ φ “ 0

(as φpKGq Ď I) and henceφ “ φ ˝ π ´ π ˝ φ @φ P J. p˚q

Let now 0 ‰ φ P J (exists since S “ hdpPq Ď hdpKGq), and let α P EndKGpKGq, so φ ˝ α P J and

φα “ φαπ ´ πφα by (˚).

Set then a :“ αp1q, b :“ φp1q, c :“ πp1q, so

ab “ αp1qφp1q “ φpαp1qq “ φpαpπp1qqq ´ πpφpαp1qqq “ cab´ abc

and it follows from Remark 23.3 that

pa, bq “ pab, 1q “ pcab, 1q ´ pabc, 1q “ pc, abq ´ pab, cq “ 0.

This is true for every α P EndKGpKGq, and hence every a P KG. Finally, as p , q is non-degenerate, wehave b “ 0 and hence φ “ 0. Contradiction!


Corollary 23.5Let S be a simple KG-module.

(a) If P is any projective KG-module, then the multiplicity of S in P{ radpPq equals the multiplicityof S in socpPq. In particular

dimK pPGq “ dimK pPGq “ dimK pP˚qG “ dimK pP˚qG .

(b) We have pPSq˚ – PS˚ .

Proof :

(a) By Theorem 23.4, the first claim holds for the PIMs of KG, hence this is also true for any finite directsum of PIMs, because taking socles and radicals commute with the direct sum by Exercise 19.5(a).Next taking S “ K yields the equalities dimK pPGq “ dimK pPGq “ dimK pP˚qG “ dimK pP˚qG .

(b) We have seen in the proof of Corollary 23.2 that pPSq˚ is the projective cover of the simple modulepsocpPSqq˚. Moreover, by Theorem 23.4

psocpPSqq˚ – pPS{ radpPSqq˚ – S˚ .

Hence pPSq˚ – PS˚ .

Finally, we see that the symmetry of the group algebra also leads us to the symmetry of the Cartanmatrix.

Theorem 23.6If S and T are simple KG-modules, then

cST ¨ dimK EndKGpSq “ cTS ¨ dimK EndKGpT q .

In particular, if K is a splitting field for G, then the Cartan matrix of KG is symmetric.

Proof : By Corollary 22.3,cST “ dimK HomKGpPS , PT q{ dimK EndKGpSq

andcTS “ dimK HomKGpPT , PSq{ dimK EndKGpT q ,

so it is enough to prove that dimK HomKGpPS , PT q “ dimK HomKGpPT , PSq.Now, by Lemma 14.2 and Lemma 13.4 we have

HomKGpPS , PT q “ HomK pPS , PT qG – ppPSq˚ bK PT qG

andHomKGpPT , PSq “ HomK pPT , PSqG – ppPT q˚ bK PSqG .

Moreover, as pPSq˚ bK PT is projective by Proposition 21.1(a), it follows from Corollary 23.5(a), that

dimK pppPSq˚ bK PT qGq “ dimK pppPSq˚ bK PT q˚qGq .

But ppPSq˚ bK PT q˚ – PS bK pPT q˚ – pPT q˚ bK PS , thus we have proved that dimK HomKGpPS , PT q “dimK HomKGpPT , PSq.Finally, if K is a splitting field for G, then by definition EndKGpSq – K – EndKGpT q, so that thedimension of both endomorphism algebras is one and we have cST “ cTS and we conclude that theCartan matrix is symmetric.


24 Representations of cyclic groups in positive characteristicWe now describe the representations of a cyclic group G :“ Zn “ xg | gn “ 1y of order n P Zě1 overa field K of positive characteristic.

Notation: Set p :“ charpK q ą 0 and write n “ pam with a P Zě0 and gcdpp,mq “ 1. More-over, we assume that K is a splitting field for G, so it follows that K contains a primitive m-th root ofunity, which we denote by ζm. This enables us to use the theory of Jordan normal forms (Linear Algebra).

Theorem 24.1There are exactly n isomorphism classes of indecomposable KZn-modules. These correspond to then matrix representations

Ri,r : G Ñ GLrpK q, g ÞÑ

»

—

—

—

—

–

ζim 1

ζim. . .. . . 1

ζim

fi

ffi

ffi

ffi

ffi

fl

p1 ď i ď m, 1 ď r ď paq.

Proof : First notice that Ri,r (1 ď i ď m, 1 ď r ď pa) defines a matrix representation of G “ Zn sinceRi,rpgqn “ Ir . Furthermore, if pe1, . . . , erq is the standard K -basis of K r , then the only Zn-invariantsubspaces are the xe1, . . . , ejyK with 1 ď j ď r. As they form a chain, Ri,r is indecomposable for all 1 ďi ď m, 1 ď r ď pa, because it cannot be written as the direct sum of two non-trivial subrepresentations.It is also clear that the Ri,r are pairwise non-equivalent, as they are uniquely determined through K -dimension and eigenvalues at evaluation in g.It remains to prove that the Ri,r (1 ď i ď m, 1 ď r ď pa) account for all the indecomposable KZn-modules.We know from the theory of Jordan normal form, that if M is a KG-module with dimK pMq “: r P Zą0,then choosing a suitable K -basis, we may assume that M corresponds to a matrix representation Rsuch that Rpgq is a block diagonal matrix where each block is a Jordan block. Assuming now that M isindecomposable, then there can be only one Jordan block. Moreover, as

Rpgqn “ Rpgnq “ Rp1Gq “ Ir ,

the eigenvalues (i.e. the diagonal entries of the Jordan blocks) can only be n-th roots of unity in K , andhence they are powers ζ im (1 ď i ď m) of ζm since charpK q “ p. Furthermore,

Rpgq “ su “ us

with s “ diagpζ im, . . . , ζ imq and u is the Jordan block with diagonal entries equal to 1, where it holds that

pu´ Irqps“ ups ´ Ir “ 0 @ ps ě r,

so u is the p-part of Rpgq and it follows that 1 ď r ď pa.

Corollary 24.2Up to isomorphism:

¨ the simple KZn-modules correspond precisely to the matrix representations Ri,1 (1 ď i ď m);

¨ the PIMs of KZn correspond precisely to the matrix representations Ri,pa (1 ď i ď m).


Proof : We can bound the number of modules in both families of modules as follows.

¨ Firstly, the matrix representations Ri,1 (1 ď i ď m) all have degree 1, hence they must be irreducibleand correspond to simple KZn-modules. It follows that there are at least m pairwise non-isomorphicsimple KZn-modules.

¨ Secondly, by Corollary 21.5, if P is a PIM of KZn, then pa | dimK pPq. Hence, up to equivalence,P corresponds to one of the matrix representations Ri,pa with 1 ď i ď m. It follows that there areat most m pairwise non-isomorphic PIMs.

However, we know from Theorem 21.2(c) that there is a bijection between the simple KZn-modules andthe PIMs of KZn. The claim follows.

Chapter 7. Indecomposable Modules

After simple and projective modules, the goal of this chapter is to understand indecomposable modules ingeneral. Apart for exceptions, the group algebra is of wild representation type, which, roughly speaking,means that it is not possible to classify the indecomposable modules over such algebras. However,representation theorists have developed tools which enable us to organise indecomposable modulesin packages parametrised by parameters that are useful enough to understand essential properties ofthese modules. In this respect, first we will generalise the idea of projective modules seen in Chapter 6by defining what is called relative projectivity. This will lead us to introduce the concepts of verticesand sources of indecomposable modules, which are two typical examples of parameters bringing ususeful information about indecomposable modules in general.

Notation: throughout this chapter, unless otherwise specified, we let G denote a finite group and K bea commutative ring. All modules over group algebras considered are assumed to be free of finite rankas K -modules.

References:





[Thé95] J. Thévenaz. G-algebras and modular representation theory. Oxford Mathematical Mono-graphs. The Clarendon Press, Oxford University Press, New York, 1995.


25 Relative projectivityThe relationship between the representations of a group and those of its subgroups are one of the mostimportant tools in representation theory of finite groups. In the context of modular representation the-ory this relationship shows itself in a refinement of the notion of projectivity, namely relative projectivity.

64


Definition 25.1Let H ď G.

(a) A KG-module M is called H-free if M – V ÒGH for some KH-module V .

(b) A KG-module M is called relatively H-projective, or simply H-projective, if it is isomorphicto a direct summand of an H-free module, i.e. if M | V ÒGH for some KH-module V .

Remark 25.2It is easy to see that H-freeness is a generalisation of freeness and relative projectivity is ageneralisation of projectivity. Indeed,

¨ freeness is the same as t1u-freeness: as KG – K ÒGt1u by Example 7, clearly pKGqn –

pK nqÒGt1u;

¨ projectivity is the same as t1u-projectivity: a KG-module is projectiveô it is a direct summandof a free KG-module ô it is a direct summand of a t1u-free KG-module ô it is relativelyt1u-projective.

To begin with, we would like to characterise relative projectivity in a similar way we characterised pro-jectivity in Proposition-Definition B.5. To reach this aim, we first take a closer look at the adjunctionbetween induction and restriction, we have seen in Theorem 15.10.

Notation 25.3Let H ď G.

(1) Let φ : U1 ÝÑ U2 be a KH-homomorphism. Then we denote by φ ÒGH the induced KG-homomorhpism

φÒGH :“ IdKG bφ : U1 ÒGH “ KG bKH U1 ÝÑ U2 Ò

GH “ KG bKH U2

x b u ÞÑ x b φpuq.

obtained by tensoring with the identity morphism on KG.

(2) Let U be a KH-module and V be a KG-module. The K -isomorphism

Φ :“ ΦU,V : HomKGpU ÒGH , V q–ÝÑ HomKHpU,V ÓGHq

andΨ :“ ΨU,V : HomKHpU,V ÓGHq

–ÝÑ HomKGpU ÒGH , V q

we obtained in Theorem 15.10 hides the fact that the induction and restriction functors IndGHand ResGH form what is called a pair of adjoint functors in category theory. More precisely,this isomorphism translates the fact that IndGH is left adjoint to ResGH .Explained in more details, there may of course be many such isomorphisms, but there is achoice which is called natural in U and V . Spelled out, this means that whenever a morphismγ P HomKHpU1, U2q is given, the diagram


HomKGpU1 ÒGH , V q HomKHpU1, V ÓGHq

HomKGpU2 ÒGH , V q HomKHpU2, V ÓGHq

ΦU1 ,V

–

ΦU2 ,V

–

pγÒGHq˚ γ˚

commutes and whenever α P HomKGpV1, V2q is given, the diagram

HomKHpU,V1 ÓGHq HomKGpU ÒGH , V1q

HomKHpU,V2 ÓGHq HomKGpU ÒGH , V2q

ΨU,V1–

α˚ α˚

ΨU,V2

–

commutes. (For the upper and lower ˚ notation, see again Proposition D.3.)Furthermore, such natural isomorphisms depend on certain distinguished homomorphismscalled the unit and the counit of the adjunction. In order to understand relative H-projectivity,we need to consider the KH-homomorphism

µ : U ÝÑ U ÒGHÓGH “à

gPrG{Hsgb U “ 1b U ‘

à

gPrG{Hs,g‰1gb U

u ÞÑ 1b u

(i.e. the natural inclusion of U into the summand 1b U) and the KG-homomorphism

ε : V ÓGHÒGH “à

gPrG{Hsgb pV ÓGHq ÝÑ V

gb v ÞÑ gv

which are, respectively, the unit and the counit of the adjunction saying that IndGH is leftadjoint to ResGH . Further, we note that for any u P U , we have ε ˝ µpuq “ εp1 b uq “ u, soε ˝ µ “ IdU and thus we deduce that:

¨ µ is a KH-section for ε ;¨ µ is injective; and¨ ε is surjective.

This allows us to construct mutually inverse natural K -isomorphisms

Φ “ ΦU,V : HomKGpU ÒGH , V q ÝÑ HomKHpU,V ÓGHq, ψ ÞÑ ψ ˝ µ ,

Ψ “ ΨU,V : HomKHpU,V ÓGHq ÝÑ HomKGpU ÒGH , V q, β ÞÑ ε ˝ β ÒGH .

as required.


Proposition 25.4 (Characterisation of relative projectivity)Let H ď G. Let U be a KG-module. Then the following are equivalent.

(a) The KG-module U is relatively H-projective.(b) If ψ : U ÝÑ W is a KG-homomorphism, φ : V � W is a sur-

jective KG-homomorphism and there exists a KH-homomorphismαH : U ÓGHÝÑ V ÓGH such that φ ˝ αH “ ψ on U ÓGH , then thereexists a KG-homomorphism αG : U ÝÑ V such that φ ˝ αG “ ψ sothat the diagram on the right commutes.

U

V W

ψö

DαG

φ

(c) Whenever φ : V � U is a surjective KG-homomorphism such that the restrictionφ : V ÓGH ÝÑ U ÓGH splits as KH-homomorphism, then φ splits as a KG-homomorhpism.

(d) The surjective KG-homomorphism

U ÓGHÒGH“ KG bKH U ÝÑ Ux b u ÞÑ xu

is split.

(e) The KG-module U is a direct summand of U ÓGHÒGH .

(f ) There exists a KG-module N such that U | K ÒGH bKN .

Proof :(a)ñ(b): First we consider the case where U “ T ÒGH is an induced module. Suppose that we have KG-

homomorphisms ψ : T ÒGH ÝÑ W and φ : V � W as shown in the diagram shown on the leftbelow. Suppose, moreover, that there exists a KH-homomorhpism αH : T ÒGHÓGHÝÑ V ÓGH such thatψ “ φ ˝ αH , that is, the diagram on the right below commutes:

T ÒGH

V W

ψ

φ

T ÒGHÓGH

V ÓGH W ÓGH

ψö

DαH

φ

Let µ : T ÝÑ T ÒGHÓGH and ε : V ÓGHÒGHÝÑ V be the unit and the counit of the adjunction of ResGH andIndGH as defined in Notation 25.3, so µ is an injective KH-homomorphism and ε is a surjective KG-homomorphism. Then, precomposing with µ, we obtain that the following triangle of KH-modulesand KH-homomorphisms commutes:

T

V ÓGH W ÓGH

ψ˝µö

αH˝µ

φ

By the naturality of Φ and Ψ from Notation 25.3, since φ : V ÝÑ W is a KG-homomorphism, wehave the following commutative diagram:


HomKHpT , V ÓGHq HomKGpT ÒGH , V q

HomKHpT ,W ÓGHq HomKGpT ÒGH ,W q

ΨT ,V

–

φ˚ φ˚

ΨT ,W

–

In other words,Ψpφ ˝ pαH ˝ µqq “ φ ˝ pΨpαH ˝ µqq .

By the commutativity of the previous triangle, the left hand side of this equation is equal toΨpψ ˝ µq “ ΨpΦpψqq “ ψ since Ψ and Φ are inverse to one another. Thus

ψ “ φ ˝ ε ˝ ppαH ˝ µqÒGHq

and so the triangle of KG-homomorphisms

T ÒGH

V W

ψö

ε˝ppαH˝µqÒGHq

φ

commutes, proving the implication for U “ T ÒGH an induced module.Now let U be any direct summand of T ÒGH . Let U ι

ÝÑ T ÒGHπÝÑ U denote the canonical inclusion and

projection. Suppose that there is a KH-homomorphism αH : U ÓGHÝÑ V ÓGH such that the diagram

U ÓGH

V ÓGH W ÓGH

ψö

DαH

φ

commutes, i.e. φ ˝ αH “ ψ on U ÓGH . Then we consider the following diagrams:

T ÒGH

V W

ψ˝π

φ

T ÒGHÓGH

V ÓGH W ÓGH

ψ˝πö

αH˝π

φ

T ÒGH

V ÓGH W

ψ˝πö

αG

φ

The middle diagram of KH-homomorphisms commutes by definition of αH , and hence by the firstpart there is a KG-homomorphism αG : T ÒGHÝÑ V such that φ ˝ αG “ ψ ˝ π, so the third diagramof KG-homomorhpisms also commutes.Now φ ˝ αG ˝ ι “ ψ ˝ π ˝ ι “ ψ, so the triangle

U

V W

ψö

αG˝ι

φ

commutes, as required.(b)ñ(c): Let φ : V � U be a surjective KG-homomorphism which is split as a KH-homomorphism, and let

αH be a KH-section for φ. Thus, we have the following commutative diagram of KH-modules:


U ÓGH

V ÓGH U ÓGH

IdUö

αH

φ

Then assuming (b) is true, there exists a KG-homomorphism αG : U ÝÑ V such that φ ˝ αG “ IdU .In particular, αG is a KG-section for φ.

(c)ñ(d): Since µ : U ÝÑ U ÓGHÒGH is a KH-section for ε : U ÓGHÒGHÝÑ U (see Notation 25.3), applyingcondition (c) yields that ε splits as a KG-homomorhpism, and hence (d) holds.

(d)ñ(e): is immediate.(e)ñ(f ): Recall that by Proposition 15.11 we have K ÒGH bKN – pK bK N ÓGHq ÒGH– N ÓGHÒGH . Thus, setting

N :“ U yields the claim.(f)ñ(a): straightforward from the fact that K ÒGH bKN – N ÓGHÒGH seen above.

Exercise 25.5Let H ď J ď G. Let U be a KG-module and let V be a KJ-module. Prove the following statements.

(a) If U is H-projective then U is J-projective.

(b) If U is a direct summand of V ÒGJ and V is H-projective, then U is H-projective.

(c) For any g P G, U is H-projective if and only if gU is gH-projective.

(d) Using part (f) of Proposition 25.4, prove that if U is H-projective and W is any KG-module,then U bK W is H-projective.

Projectivity relative to a subgroup can be generalised as follows to projectivity relative to a KG-module:

Remark 25.6 (Projectivity relative to KG-modules)

(a) Let V be a KG-module. A KG-module M is termed projective relative to the module V orrelatively V -projective, or simply V -projective if there exists a KG-module N such that M isisomorphic to a direct summand of V bK N , i.e. M | V bK N .We let ProjpV q denote the class of all V -projective KG-modules.

(b) Proposition 25.4(f) shows that projectivity relative to a subgroup H ď G is in fact projectivityrelative to the KG-module V :“ K ÒGH .

Note that the concept of projectivity relative to a subgroup is proper to the group algebra, but theconcept of projectivity relative to a module is not and makes sense in general over algebras/rings.

The following exercise provides us with some elementary properties of projectivity relative to a module,which also hold for projectivity relative to a subgroup, by part (b) of the remark.


Exercise 25.7Assume K is a field of characteristc p ą 0 and let A,B, C, U, V be KG-modules. Prove that:

(a) Any direct summand of a V -projective KG-module is V -projective;

(b) If U P ProjpV q, then ProjpUq Ď ProjpV q;

(c) If p - dimK pV q then any KG-module is V -projective;

(d) ProjpV q “ ProjpV ˚q;

(e) ProjpU ‘ V q “ ProjpUq ‘ ProjpV q;

(f ) ProjpUq X ProjpV q “ ProjpU bK V q;

(g) ProjpÀn

j“1 V q “ ProjpV q “ ProjpÂm

j“1 V q @m,n P Zą0;

(h) C – A‘ B is V -projective if and only if both A and B are V -projective;

(i) ProjpV q “ ProjpV ˚ bK V q.

Hint: you may want to use Lemma 13.8 and Exercise 4(c) on Sheet 3. Proceed in the given order.

After this small parenthesis on projectivity relative to modules, we come back to projectivity relativeto subgroups. We investigate further what information this concept brings to the understanding ofindecomposable KG-modules in general.

Next we see that any indecomposable KG-module can be seen as a relatively projective module withrespect to some subgroup of G.

Theorem 25.8Let H ď G.

(a) If |G : H| is invertible in K , then every KG-module is H-projective.

(b) In particular, if K is a field of characteristic p ą 0 and H contains a Sylow p-subgroup of G,then every KG-module is H-projective.

Proof : (a) Let V be a KG-module. To prove that V is H-projective, we prove that V satisfies The-orem 25.4(c). So let φ : U � V be a surjective KG-homomorphism which splits as a KH-homomorphism. We need to prove that φ splits as a KG-homomorphism.So let σ : V ÝÑ U be a KH-linear section for φ and set

rσ : V ÝÑ Uv ÞÑ 1

|G:H|ř

gPrG{Hs g´1σpgvq.

We may divide by |G : H| since |G : H| P Kˆ and clearly rσ is well-defined. Now, if g1 P G andv P V , then

rσpg1vq “ 1|G : H|

ÿ

gPrG{Hsg´1σpgg1vq “ g1 1

|G : H|ÿ

gPrG{Hspgg1q´1σpgg1vq “ g1rσpvq


and

φrσpvq “ 1|G : H|

ÿ

gPGφ`

g´1σpgvq˘ φ KG-lin.

“1

|G : H|ÿ

gPGg´1φσpgvq “ 1

|G : H|ÿ

gPGg´1gv “ v

where the last-but-one equality holds because φσ “ IdV . Thus rσ is a KG-linear section for φ.(b) follows immediately from (a). Indeed, if P P SylppGq and H Ě P , then p - |G : H|, so |G : H| P Kˆ.

Considering the case H “ t1u shows that the previous Theorem is in some sense a generalisation ofMaschke’s Theorem (Theorem 11.1).

Remark 25.9Assume that K is a field of characteristic p ą 0 and H “ t1u is the trivial subgroup. If H containsa Sylow p-subgroup of G then the Sylow p-subgroups of G are trivial, so p - |G|. The theoremthen says that all KG-modules are t1u-projective and hence projective.We know this already, however! If p - |G| then KG is semisimple by Maschke’s Theorem (Theo-rem 11.1), and so all KG-modules are projective by Example 10(d).

Corollary 25.10Let H ď G and suppose that |G : H| is invertible in K . Then a KG-module U is projective if andonly if U ÓGH is projective.

Again, this holds in particular if K is a field of characteristic p ě 0 and H contains a Sylow p-subgroupof G.

Proof : The necessary condition is given by Proposition 21.1(b). To prove the sufficient condition, supposethat U ÓGH is projective. Then, on the one hand,

U ÓGH | pKHqn for some n P Zą0 .

On the other hand, U is H-projective by Theorem 25.8, and it follows from Proposition 25.4(e) that

U | U ÓGHÒGH .

HenceU | U ÓGHÒGH | pKHqn ÒGH – pKGqn ,

so U is projective.

26 Vertices and sourcesFor the remainder of this chapter, we assume that K isa field with charpK q “: p ą 0.

As said before, we now want to explain some techniques that are available to understand indecompos-able modules better. Vertices and sources are two parameters making this possible.


Theorem 26.1Let U be an indecomposable KG-module.

(a) There is a unique conjugacy class of subgroups Q of G which are minimal subject to theproperty that U is Q-projective.

(b) Let Q be a minimal subgroup of G such that U is Q-projective. Then, there exists an inde-composable KQ-module T which is unique, up to conjugacy by elements of NGpQq, such thatU is a direct summand of T ÒGQ . Such a KQ-module T is necessarily a direct summand ofU ÓGQ .

Proof :(a) Suppose that U is both H- and K -projective for subgroups H and K of G. Then, by Proposi-

tion 25.4(e),U | U ÓGHÒGH

and the latter module is K -projective. Hence U is also a direct summand of U ÓGHÒGHÓGKÒGK . By theMackey formula and transitivity of induction and restriction, it follows that

U ÓGHÒGHÓGKÒGK “´

`

U ÓGH˘

ÒGHÓGK

¯

ÒGK

“

´

à

gPrKzG{Hs

` gÙ ÓGH˘

ÓgHKX gH

˘

ÒKKX gH

¯

ÒGK

“à

gPrKzG{Hs

` gU ÓGKX gH˘

ÒGKX gH .

Therefore, by the Krull-Schmidt Theorem, there exists g P G such that U is a direct summand ofa module induced from K X gH , and hence U is K X gH-projective. Now, if both K and H areminimal such that U is projective relative to these subgroups, then K X gH “ K . Thus, K Ď gHand H Ď g´1K , hence H and K are G-conjugate.

(b) Let Q be a minimal subgroup relative to which U is projective. Then U | U ÓGQÒGQ by Proposi-tion 25.4(e) so it is a direct summand of T ÒGQ for some indecomposable direct summand T of U ÓGQ .If T 1 is another indecomposable KQ-module such that U | T 1 ÒGQ , then T | T 1 ÒGQÓGQ . Mackey’sformula says that

T 1 ÒGQÓGQ “à

gPrQzG{Qsp gT 1 Ó

gQQX gQqÒ

QQX gQ ,

hence, again by the Krull-Schmidt Theorem, there exists g P G such that

T | p gT 1 ÓgQQX gQqÒ

QQX gQ

and therefore T is Q X gQ-projective, and hence so is U . Since Q is a minimal subgroup relativeto which U is projective, Q “ Q X gQ and hence g P NGpQq. It follows that T is actually a directsummand of gT 1, for this g P G. Since T and T 1 are indecomposable, however, this means thatT “ gT 1, so T is unique up to conjugacy by elements of NGpQq.Now T “ gT 1 is an idecomposable direct summand of U ÓGQ by definition, so T 1 “ g´1T is a directsummand of p g´1UqÓGQ . However, U – g´1U as KG-modules, so this means that T 1 is also a directsummand of U ÓGQ .

This characterisation leads us to the following definition:


Definition 26.2Let U be an indecomposable KG-module.

(a) A vertex of U is a minimal subgroup Q of G such that U is relatively Q-projective.The set of all vertices of U is denoted by vtxpUq.

(b) Given a vertex Q of U , a KQ-source, or simply a source of U is a KQ-module T such thatU | T ÒGQ .

Remark 26.3

(a) Conceptually, the closer the vertices of a module are to the trivial subgroup, the closer thismodule is to being projective: a KG-module U with trivial vertex is t1u-projective and henceprojective.

(b) A vertex Q of an indecomposable KG-module U is not uniquely defined, in general. However,the vertices of U are unique up to G-conjugacy, so in particular are all isomorphic. For thisreason, in general, one (i.e. you!) should never talk about the vertex of a module (of course,unless a vertex has been fixed). We either say that Q is a vertex of U , or talk about thevertices of U . (Unfortunately many textbooks/articles by non-experts are very sloppy withthis terminology, inducing errors.)

(c) For a fixed vertex Q of U , a source of U is defined up to conjugacy by elements of NGpQq.

To begin with, we have the following important restriction on the structure of the vertices.

Proposition 26.4The vertices of an indecomposable KG-module are p-subgroups of G.

Proof : By Theorem 25.8, we know that every KG-module is projective relative to a Sylow p-subgroupof G. Therefore, by minimality, vertices are contained in Sylow p-subgroups, and hence are themselvesp-groups.

Warning: vertices and sources are very useful theoretical tools in general, but extremely difficult tocompute concretely.

We show here how to compute the vertices and sources of the trivial module.

Example 11The vertices of the trivial KG-module K are the Sylow p-subgroups of G, i.e. vtxpK q “ SylppGq,and all sources are trivial.

To establish this fact, we need the following indecomposability property:

Claim: If P is a p-group and H ď P , then K ÒPH is an indecomposable KP-module.Indeed: First recall that as P is a p-group, the only simple KP-module is the trivial module. Hencethe socle of K ÒPH is a direct sum of trivial submodules. This together with Frobenius reciprocity


yields

dimK socpK ÒPHq “ dimK HomKPpK,K ÒPHq “ dimK HomKHpK ÓPH , K q “ dimK HomKHpK,K q “ 1 .

If K ÒPH were decomposable, then so would be its socle: clearly, if K ÒPH “ U ‘ V for some KP-modules U,V ‰ 0, then

dimK socpK ÒPHq “ dimK psocpUq ‘ socpV qq “ dimK socpUq ` dimK socpV q ě 1` 1 “ 2 .

A contradiction! Therefore K ÒPH is indecomposable.

Now, let Q P vtxpK q and let P P SylppGq such that P ě Q. Since K is Q-projective,

K | K ÓGQÒGQ “ K ÒGQ

soK ÓGP | K ÒGQÓGP “

à

gPrPzG{QsK ÒPPX gQ

by the Mackey formula, and hence, by the Krull-Schmidt Theorem, is a direct summand of K ÒPPX gQfor some g P G, which is indecomposable by the Claim. Thus

K “ K ÓGP “ K ÒPPX gQ

and hence P X gQ “ P , so gQ “ P . Therefore, Q is a Sylow p-subgroup of G and it followsfrom Theorem 26.1(a), that vtxpK q “ SylppGq. Finally, it is clear that the trivial KQ-module is aKQ-source, and hence all sources are trivial.

27 The Green correspondenceThe Green correspondence is a correspondence which relates the indecomposable KG-modules witha fixed vertex with the indecomposable KL-modules with the same vertex for well-chosen subgroupsL ď G. It is used to reduce questions about indecomposable modules to a situation where a vertex of thegiven indecomposable module is a normal subgroup. This technique is very useful in many situations.In fact, many properties in modular representation theory are believed to be determined by normalisersof p-subgroups.

Lemma 27.1Let Q ď G be a p-subgroup and let L ď G.

(a) If U is an indecomposable KG-module with vertex Q and L ě Q, then there exists an inde-composable direct summand of U ÓGL with vertex Q.

(b) If L ě NGpQq, then the following assertions hold.

(i) If V is an indecomposable KL-module with vertex Q and U is a direct summand of V ÒGLsuch that V | U ÓGL , then Q is also a vertex of U .

(ii) If V is an indecomposable KL-module which is Q-projective and there exists an inde-composable direct summand U of V ÒGL with vertex Q, then Q is also a vertex of V .

Proof : Exercise, Sheet 5.


Theorem 27.2 (Green Correspondence)Let Q be a p-subgroup of G and let L be a subgroup of G containing NGpQq.

(a) If U is an indecomposable KG-module with vertex Q, then

U ÓGL “ fpUq ‘ X

where fpUq is the unique indecomposable direct summand of U ÓGL with vertex Q and everydirect summand of X is LX xQ-projective for some x P GzL.

(b) If V is an indecomposable KL-module with vertex Q, then

V ÒGL “ gpV q ‘ Y

where gpV q is unique indecomposable direct summand of V ÒGL with vertex Q and every directsummand of Y is Q X xQ-projective for some x P GzL.

(c) With the notation of (a) and (b), we then have gpfpUqq – U and fpgpV qq – V . In other words,f and g define a bijection

!

isomorphism classes of indecomposableKG-modules with vertex Q

)

„ÐÑ

!

isomorphism classes of indecomposableKL-modules with vertex Q

)

U ÞÑ fpUq

gpV q Ð [ V .

Moreover, corresponding modules have a source in common.

Terminology: fpUq is called the KL-Green correspondent of U (or simply the Green correspondent)and gpV q is called the KG-Green correspondent of V (or simply the Green correspondent of V ).

Warning! In the Green correspondence it is essential that a vertex Q is fixed and not considered upto conjugation, because the G-conjugacy class of Q and the L-conjugacy class of Q do not coincide ingeneral.

Proof : We first note some properties of the subgroups Q X xQ and LX xQ for x P GzL :

p˚q Since NGpQq ď L, x does not normalise Q and hence Q X xQ is a proper subgroup of Q.p˚˚q LX xQ may be of the same order as Q, in which case LX xQ “ xQ.

p˚ ˚ ˚q Suppose that L X xQ is conjugate to Q in L, i.e. there exists z P L such that L X xQ “ zQ. ThenxQ “ zQ so z´1xQ “ Q and hence z´1x P NGpQq ď L. Therefore x P zL “ L. This contradictsx P GzL. Therefore LX xQ is never conjugate to Q in L.

(b) Let V be an indecomposable KL-module with vertex Q.

Claim. The KL-module V ÒGL ÓGL has a unique direct summand with vertex Q, and all other directsummands are projective relative to subgroups of the form LX xQ with x P GzL.Pf of the Claim: Let T be a KQ-source for V . Then, we may write T ÒLQ“ V ‘ Z for someKL-module Z . Moreover, there exist KL-modules V 1 and Z 1 such that V ÒGL ÓGL“ V ‘ V 1 andZ ÒGL ÓGL“ Z ‘ Z 1. Then, on the one hand we have

T ÒGQÓGL “ pV ‘ Z qÒGL ÓGL “ V ÒGL ÓGL ‘Z ÒGL ÓGL “ V ‘ V 1 ‘ Z ‘ Z 1 .


On the other hand, by the Mackey formula we also have

T ÒGQÓGL –à

xPrLzG{Qsp xT Ó

xQLXxQqÒ

LLXxQ

“ T ÒLQ ‘à

xPrLzG{QsxRL

p xT ÓxQLXxQqÒ

LLXxQ

“ V ‘ Z ‘à

xPrLzG{QsxRL

p xT ÓxQLXxQqÒ

LLXxQ .

Therefore, by the Krull-Schmidt Theorem, we have

V 1 ‘ Z 1 –à

xPrLzG{QsxRL

p xT ÓxQLXxQqÒ

LLXxQ

where, clearly, all direct summands are L X xQ-projective for some x R L. It follows that V is theunique direct summand of V ÒGL ÓGL“ V ‘ V 1 with vertex Q, because all the direct summands in V 1are projective relative to subgroups of the form L X xQ with x R L and so are not Q-projective byp˚ ˚ ˚q. This proves the Claim.Now, write V ÒGL as a direct sum of indecomposable KG-modules and pick a direct summand Usuch that V | U ÓGL . By Lemma 27.1(b), since Q is a vertex of V , Q is also a vertex of U . ThereforeV ÒGL has at least one direct summand with vertex Q. To prove its uniqueness, assume U 1 is anotherindecomposable direct summand of V ÒGL . Then

V ÒGL“ U ‘ U 1 ‘ X

for some KG-module X , so in the notation of the claim,

V ‘ V 1 “ U ÓGL ‘U 1 ÓGL ‘X ÓGL .

As V | U ÓGL , by the Krull-Schmidt-Theorem, U 1 ÓGL | V 1 and hence every indecomposable directsummand of U 1 ÓGL is LXyQ-projective for some y R L by the proof of the Claim. Now since V | T ÒLQand U 1 | V ÒGL it follows that

U 1 | T ÒLQÒGL“ T ÒGQ .Thus U 1 is Q-projective and therefore has a vertex Q1 contained in Q.It remains to prove that Q1 ň Q. So let S be a KQ1-source of U 1. Then S | U 1 ÓGQ1 by The-orem 26.1(b). Since Q1 ď L, U 1 ÓGQ1“ U 1 ÓGL ÓLQ1 and hence S is a direct summand of Y ÓLQ1 forsome indecomposable direct summand Y of U’ ÓGL . It follows from Lemma 27.1 that Q1 is also avertex of Y . But the indecomposable direct summands of U 1 ÓGL are all L X yQ-projective for somey R L. Therefore one of the subgroups L X yQ with y R L contains an L-conjugate of Q1. I.e.zQ1 Ď LX yQ for some z P L. Hence Q1 Ď z´1yQ where z´1y R L and so there exists x P GzL suchthat Q1 Ď Q X xQ ň Q by p˚q. Therefore, we may set gpV q :“ U and the assertion follows.

(a) Suppose now that U is an indecomposable KG-module with vertex Q and let T be a KQ-sourceof U . Then U is a direct summand of T ÒGQ“ T ÒLQÒGL , so there is an indecomposable direct summandV of T ÒLQ such that U | V ÒGL . This means that V is Q-projective, and so Q is a vertex of V byLemma 27.1(b).Now, by Lemma 27.1(a), there exists an indecomposable direct summand Y of U ÓGL with vertex Q.But U ÓGL | V ÒGL ÓGL and the Claim says that the only direct summand of V ÒGL ÓGL with vertex Qis V . Therefore Y – V and the remaining direct summands of U ÓGL are LX xQ-projective for somex P GzL. This proves part (a).

(c) The claim follows immediately from parts (a) and (b) and the facts that U | U ÓGL ÒGL and V | V ÒGL ÓGL .


Exercise 27.3

(a) Verify that modules corresponding to each other via the Green correspondence have a sourcein common.

(b) Prove that the Green correspondent of the trivial module is the trivial module.

28 p-permutation modulesRecall from Example 4(b) that any finite G-set X gives rise to a K -representation ρX of G. The KG-module corresponding to ρX through Proposition 10.3 admits X as K -basis, hence it is standard todenote this module by KX and it is called the permutation KG-module on X . Thus, we may pose thefollowing definition.

Definition 28.1 (Permutation module)A KG-module is called a permutation KG-module if it admits a K -basis X which is invariant underthe action of G. We denote this module by KX .

(Note: it is clear that the basis X is then a finite G-set.)

Permutation KG-modules and, in particular, their indecomposable direct summands have remarkableproperties, which we investigate in this section.

Remark 28.2First, we describe permutation modules and some of their properties more precisely.

If KX is a permutation KG-module on X , then a decomposition of the basis X as a disjoint unionof G-orbits, say X “

Ůni“1 Xi , yields a direct sum decomposition of KX as a KG-module as

KX “nà

i“1KXi .

Thus, without loss of generality, we can assume that X is a transitive G-set, in which case

KX – K ÒGH

where H :“ StabGpxq, the stabiliser in G of some x P X . Indeed, clearly we have a direct sumdecomposition as a K -vector space

KX “à

gPrG{HsKgx

and G acts transitively on the summands, so that KX “ K ÒGH .

It follows that an arbitrary permutation KG-module is isomorphic to a direct sum of KG-modules ofthe form K ÒGH for various H ď G.

Notice that, conversely, an induced module of the form K ÒGH (H ď G) is always a permutationKG-module. Indeed, as K ÒGH “ KGbKH K “

À

gPrG{Hs gbK as K -vector space, it has on obvious


G-invariant K -basis given by the set

tgb 1K | g P rG{Hsu .

In fact, more generally if H ď G and KX is a permutation KH-module on X , then KX ÒGH is apermutation KG-module with G-invariant K -basis tg b x | g P rG{Hs, x P Xu . In other words,induction preserves permutation modules.

Exercise 28.3Prove that direct sums, restriction, inflation and conjugation also preserve permutation modules.

Next, we investigate the indecomposable direct summands of the permutation KG-modules. In order tounderstand the indecomposable ones, we are going to prove that they all have a trivial source and wewill apply the Green correspondence to see that, up to isomorphism, there are only a finite number ofthem.

Definition 28.4 (trivial source module)A KG-module is called a trivial source KG-module if it is a finite direct sum of KG-modules witha trivial source K .

Warning: Some texts (books/articles/. . . ) require that a trivial source module is indecomposable, othersdo not.

Proposition-Definition 28.5 (p-permutation module)Let M be a KG-module and let P P SylppGq. Then, the following conditions are equivalent:

(a) M ÓGQ is a permutation KQ-module for each p-subgroup Q ď G;

(b) M ÓGP is a permutation KP-module;

(c) M has a K -basis which is invariant under the action of P;

(d) M is isomorphic to a direct summand of a permutation KG-module;

(e) M is a trivial source KG-module.

If M fulfils one of these equivalent conditions, then it is called a p-permutation KG-module.

Note. In fact p-permutation KG-modules and trivial source KG-modules are two different pieces ofterminology for the same concept. French/German speaking authors tend to favour the use of the termi-nology p-permutation module (and reserve the terminology trivial source module for an indecomposablemodule with a trivial source), whereas English speaking authors tend to favour the use of the terminol-ogy trivial source module.

Proof :

(a)ô(b): It is obvious that (a) implies (b). For the sufficient condition, notice that for each g P G, we haveM ÓGgP – gpM ÓGPq. Therefore, as by Exercise 28.3 restriction and conjugation preserve permutationmodules, requiring that M ÓGP is a permutation KP-module implies that M ÓGQ is a permutation


KQ-module for each p-subgroup Q ď G. (Because any p-subgroup Q of G is contained in a Sylowp-subgroup and these are all G-conjugate by the Sylow theorems.)

(b)ô(c): is obvious by the definition of a permutation KP-module.(b)ñ(e): Claim: If L is a KG-module satisfying (b), then so does any direct summand of L.

Proof of the Claim: By Remark 28.2, if LÓGP is a permutation KP-module, then there exist n P Zně1and subgroups Qi ď G (1 ď i ď n) such that

LÓGP–nà

i“1K ÒPQi ,

where each K ÒPQi is indecomposable by the Claim in Example 11. Therefore, by the Krull–Schmidttheorem, if N | L, then N ÓGP is isomorphic to the direct sum of some of the factors, hence is againa permutation KP-module (by Remark 28.2) and so N satisfies (b) as well, as required.Now, if M satisfies (b), then by the Claim we can assume w.l.o.g. that M is indecomposable. Let Qbe a vertex of M . Then M | M ÓGQÒGQ by Q-projectivity. Since M ÓGQ is a permutation KQ-moduleby (a) (ô (b)), again by Remark 28.2, there exist n P Zě0 and subgroups Ri ď Q (1 ď i ď n) suchthat

M ÓGQ –nà

i“1KÒQRi .

Inducing this module to G again and using the Krull-Schmidt theorem, we deduce that M , beingindecomposable, is isomorphic to a direct summand of K ÒGRi for some 1 ď i ď n. By minimalityof the vertex Q, it follows that Ri “ Q and that the trivial KQ-module K must be a source of M ,proving that M is a trivial source KG-module.

(e)ñ(d): If L is an indecomposable trivial source module, say with vertex Q ď G, then by definition of asource, L | K ÒGQ . This implies (d) as K ÒGQ is a permutation KG-module by Remark 28.2 and anyfinite direct sum of permutation KG-module is again permutation.

(d)ñ(b): Assume that M | Z , where Z is a permutation KG-module. Then M ÓGP |Z ÓGP , where Z ÓGP is againa permutation KP-module by Exercise 28.3. Thus, it follows from the Claim in (b)ñ(e) (see alsothe scholium below) that M ÓGP is a permutation KP-module, as required.

The Claim in (b)ñ(e) can be formulated as the following result.

Scholium 28.6If M is a p-permutation KG-module, then any direct summand of M is again a p-permutation KG-module. In particular, if G is a p-group, then any direct summand of a permutation KG-module is apermutation KG-module and so, in this case, any p-permutation module is a permutation module.

Exercise 28.7Prove that p-permutation modules are preserved by the following operations: direct sums, tensorproducts, restriction, inflation, conjugation, induction.

Example 12It is clear that any projective KG-module is a p-permutation KG-module. Also, the PIMs of KG areprecisely the KG-modules with vertex t1u and trivial source.

Generalising this example, we can characterise the indecomposable p-permutation KG-modules with agiven vertex Q ď G as described below.


Example 13

(1) If M is an indecomposable p-permutation KG-module with vertex Q ď G, then Q acts triv-ially on the KNGpQq-Green correspondent fpMq of M . Thus fpMq can be viewed as aK rNGpQq{Qs-module. As such, fpMq is indecomposable and projective.

(2) Conversely, if N is a projective indecomposable K rNGpQq{Qs-module, then InfNGpQqNGpQq{QpNq is

an indecomposable KNGpQq-module with vertex Q and trivial source. Its KG-Green corre-spondent is then also an indecomposable KG-module with vertex Q and trivial source, henceis an indecomposable p-permutation KG-module

(3) In this way we obtain a bijection!

isomorphism classes of indecomposablep-permutation KG-modules with vertex Q

)

„ÐÑ

!

isomorphism classes of projectiveindecomposable K rNGpQq{Qs-modules

)

.

29 Green’s indecomposability theoremTo finish our analysis of indecomposable KG-modules we mention without proof an important indecom-posability criterion due to J. A. Green (1959). The proof is rather involved and goes beyond the scopeof the techniques we have developed so far.

Theorem 29.1 (Green’s indecomposability criterion, 1959)Assume that K is an algebraically closed field. Let H ď G be a subnormal subgroup of G of indexa power of p and let M be an indecomposable KH-module. Then M ÒGH is an indecomposableKG-module.

Proof : Without proof in this lecture. See [Thé95, (23.6) Corollary].

Remark 29.2Green’s indecomposability criterion remains true over an arbitrary field of characteristic p, providedwe replace indecomposability with absolute indecomposability. (A KG-module M is called abso-lutely indecomposable iff its endomorphism algebra EndKGpMq is a split local algebra, that is, ifEndKGpMq{JpEndKGpMqq – K .)

Corollary 29.3Assume that K is an algebraically closed field. If P is a p-group, Q ď P andM is an indecomposableKQ-module, then M ÒPQ is an indecomposable KP-module.

Proof : By the Sylow theory, since P is a p-group, any subgroup Q ď P can be plugged in a subnormalseries where each quotient is cyclic of order p, hence is a subnormal subgroup of P . Therefore, the claimfollows immediately from Green’s indecomposability criterion.

Notice that, in Example 11, we have proved the latter result in the particular case that M “ K is thetrivial KQ-module using simple arguments.


Exercise 29.4Assume that K is an algebraically closed field and let M be an indecomposable KG-module.

(a) Let Q P vtxpMq and let P P SylppGq such that Q ď P . Prove that |P : Q|ˇ

ˇ dimK pMq .

(b) Prove that if dimK pMq is coprime to p, then vtxpMq “ SylppGq.

Chapter 8. p-Modular Systems

R. Brauer started in the late 1920’s a systematic investigation of group representations over fields ofpositive characteristic. In order to relate group representations over fields of positive characteristic tocharacter theory in characteristic zero, Brauer worked with a triple of rings pF,O, kq, called a p-modularsystem, and consisting of a complete discrete valuation ring O with a residue field k :“ O{JpOq of primecharacteristic p and fraction field F :“ FracpOq of characteristic zero. The present chapter contains ashort introduction to these concepts. We will use p-modular systems and Brauer ’s reciprocity theoremin the subsequent chapters to gain information about kG and its modules (which is/are extremelycomplicated) from the group algebra FG, which is semisimple and therefore much better understood,via the group algebra OG. This explains why we considered arbitrary associative rings (resp. fields) Kin the previous chapters rather than immediately focusing on fields of positive characteristic.Notation. Throughout this chapter, unless otherwise specified, we let p be a prime number and Gdenote a finite group. All modules are assumed to be finitely generated and free as ΛG-modules forany Λ P tF,O, ku.

References:


[Lin18] M. Linckelmann. The block theory of finite group algebras. Vol. I. Vol. 91. London Mathemat-ical Society Student Texts. Cambridge University Press, Cambridge, 2018.


[Ser68] Jean-Pierre Serre. Corps locaux. Deuxième édition, Publications de l’Université de Nancago,No. VIII. Hermann, Paris, 1968.

[Thé95] J. Thévenaz. G-algebras and modular representation theory. Oxford Mathematical Mono-graphs. The Clarendon Press, Oxford University Press, New York, 1995.


30 Complete discrete valuation ringsIn this section we review some number-theoretic results without formal proofs. I refer you to yournumber theory lectures for the details, or to [Ser68; Lin18; Thé95]. Important for the sequel is to keepthe definitions and the examples in mind.

82


To begin with, recall from Chapter 1, §5, that a commutative ring O is local iff OzOˆ “ JpOq, i.e. JpOqis the unique maximal ideal of O. Moreover, by the commutativity assumption this happens if and onlyif O{JpOq is a field. In such a situation, we write k :“ O{JpOq and call this field the residue field ofthe local ring O. To ease up notation, we will often write p :“ JpOq. This is because our aim is asituation in which the residue field is a field of positive characteristic p.

Definition 30.1 (Reduction modulo p)Let O be a local commutative ring with unique maximal ideal p :“ JpOq and residue field k :“ O{p.Let M,N be finitely generated O-modules, and let f : M ÝÑ N be an O-linear map.

(a) The k-module M :“ M{pM – k bO M is called the reduction modulo p of M .

(b) The induced k-linear map f : M ÝÑ N,m`pM ÞÑ fpmq`pN is called the reduction modulo pof f .

Exercise 30.2Let O be a local commutative ring with unique maximal ideal p :“ JpOq and residue field k :“ O{JpOq.

(a) Let M,N be finitely generated free O-modules.

(i) Let f : M ÝÑ N be an O-linear map and f : M ÝÑ N its reduction modulo p. Provethat if f is surjective (resp. an isomorphism), then f is surjective (resp. an isomorphism).

(ii) Prove that if elements x1, . . . , xn P M (n P Zě1) are such that their images x1, . . . x2 P Mform a k-basis of M , then tx1, . . . , xnu is an O-basis of M .In particular, dimkpMq “ rkOpMq.

Deduce that any direct summand of a finitely generated free O-module is free.

(b) Prove that if M is a finitely generated O-module, then the following conditions are equivalent:

(i) M is projective;(ii) M is free.

Moreover, if O is also a PID, then (i) and (ii) are also equivalent to:

(iii) M is torsion-free.

[Hint: Use Nakayama’s Lemma.]

Definition 30.3A commutative ring O is called a discrete valuation ring if O is a local principal ideal domain suchthat JpOq ‰ 0.

Example 14Let p be a prime number. We have already seen in Example 1(b) that the ring Zppq :“ tab P Q | p - buis commutative local with unique maximal ideal

JpZppqq “ tab P Zppq | p | au “ pZppq .


It follows easily that Zppq is a PID (every non-zero ideal in Zppq is of the form pnZppq for someinteger n P Zě0), hence a discrete valuation ring.

In fact this example is a special case of a more general construction principle for discrete valuationrings, which consists in taking O :“ Rp, where R is the ring of algebraic integers of an algebraicnumber field and where Rp is the localisation of R at a non-zero prime ideal p in R .

Remark 30.4There is in fact a link between Definition 30.3 and the theory of valuations explaining the termi-nology discrete valuation ring, provided by the following result. (Not difficult to prove!)Theorem 30.5

Let O be a discrete valuation ring and let π P O such that JpOq “ πO. Then:

(a) For every a P Ozt0u there is a unique maximal (non-negative) integer νpaq such thata P πνpaqO.

(b) For any a, b P Ozt0u we have νpabq “ νpaq ` νpbq and νpa` bq ě mintνpaq, νpbqu.

(c) Every non-zero ideal in O is of the form πnO for some unique integer n P Zě0.

The map ν : Ozt0u ÝÑ Z defined in this way is called the valuation of the discrete valuationring O.

One can use valuations to give an alternative definition of valuation rings. Suppose that F is afield and that ν : Fˆ ÝÑ Z is a surjective map satisfying

¨ νpabq “ νpaq ` νpbq (so ν is a group homomorphism ñ νp1q “ 0 and νpa´1q “ ´νpaq), and

¨ νpa` bq ě mintνpaq, νpbqu,

for all a, b P F , and we set for notational convenience νp0q “ 8. Then, the set

O :“ ta P F | νpaq ě 0u

is a discrete valuation ring, and F “ FracpOq is the fraction field of O. Clearly, the unique maximalideal in O is

JpOq “ ta P F | νpaq ě 1u “ OzOˆ .

Taking for π any element in O such that νpπq “ 1, one easily checks that O has the propertiesstated in the theorem above.

A valuation induces a metric, and hence a topology. For the purpose of representation theory of finitegroups, we will need to focus on the situation in which this topology is complete. This can be expressedalgebraically as follows.

Definition 30.6 (Complete discrete valuation ring)Let O be a discrete valuation ring with unique maximal ideal p :“ JpOq.

(a) A sequence pamqmě1 of elements of O is called a Cauchy sequence if for every integer b ě 1,there exists an integer N ě 1 such that am ´ an P JpOqb for all m,n ě N .


(b) O is called complete if for every Cauchy sequence pamqmě1 Ď O there is a P O such that forany integer b ě 1 there exists an integer N ě 1 such that aám P JpOqb for all m ě N . (Inthis case, a is a limit of the Cauchy sequence pamqmě1 .) It is also said that O is completewith respect to the p-adic topology.

Remark 30.7The previous definition can be generalised to a finitely generated O-algebra A. Moreover, one canprove that A is complete in the JpAq-adic topology if O is complete in the JpOq-adic topology.

Example 15Let p be a prime number. The discrete valuation ring Zppq is not complete. However, its completion,the ring of p-adic integers, that is,

Zppq “ Zp “

#

8ÿ

i“0aipi | ai P t0, . . . , p´ 1u @ i ě 0

+

,

is a complete discrete valuation ring. Its field of fractions is FracpZpq “ Qp, i.e. the field of p-adicintegers, JpZpq “ pZp and the residue field is Zp{pZp – Fp.

Finally we mention without proof a very useful consequence of Hensel’s Lemma.

Corollary 30.8Let O be a complete discrete valuation ring with unique maximal ideal p :“ JpOq and residue fieldk :“ O{p of prime characteristic p, and let m P Zě1 be coprime to p. Then, for any m-th root ofunity ζ P k , there exists a unique m-th root of unity µ P O such that µ “ ζ .

31 Splitting p-modular systemsIn order to relate group representations over a field of positive characteristic to character theory incharacter zero, Brauer chose a p-modular system, which is defined as follows:

Definition 31.1 (p-modular systems)Let p be a prime number.

(a) A triple of rings pF,O, kq is called a p-modular system if:

(1) O is a complete discrete valuation ring of characteristic zero,(2) F “ FracpOq is the field of fractions of O (also of characteristic zero), and(3) k “ O{JpOq is the residue field of O and has characteristic p.

(b) If G is a finite group, then a p-modular system pF,O, kq is a called a splitting p-modularsystem for G, if both F and k are splitting fields for G.


It is often helpful to visualise p-modular systems and the condition on the characteristic of the ringsinvolved through the following commutative diagram of rings and ring homomorphisms:

Q Z Fp

F O k

where the hook arrows are the canonical inclusions and the two-head arrows the quotient morphisms.Clearly, these morphisms also extend naturally to ring homomorphisms

FG OG kG

between the corresponding group algebras (each mapping an element g P G to itself).

Example 16One usually works with a splitting p-modular system for all subgroups of G, because it allows usavoid problems with field extensions. By a theorem of Brauer on splitting fields such a p-modularsystem can always be obtained by adjoining a primitive m-th root of unity to Qp, where m is theexponent of G. (Notice that this extension is unique.) So we may as well assume that our situationis as given in the following commutative diagram:

Qp Zp Fp

F O k

More generally, we have the following result, which we mention without proof. The proof can be foundin [CR90, §17A].

Theorem 31.2Let pF,O, kq be a p-modular system. Let G be a finite group of exponent m. Then the followingassertions hold.

(a) The field F contains all m-th roots of unity if and only if F contains the cyclotomic field ofm-th roots of unity;

(b) If F contains all m-th roots of unity, then so does k and F and k are splitting fields for Gand all its subgroups.

Remark 31.3If pF,O, kq is a p-modular system, then it is not possible to have F and k algebraically closed,while assuming O is complete. (Depending on your knowledge on valuation rings, you can try toprove this as an exercise!)

Let us now investigate changes of the coefficients given in the setting of a p-modular system for groupalgebras involved.


Definition 31.4Let O be a commutative local ring. A finitely generated OG-module L is called an OG-lattice if itis free (= projective) as an O-module.

Remark 31.5 (Changes of the coefficients)Let pF,O, kq be a p-modular system and write p :“ JpOq. If L is an OG-module, then:

¨ setting LF :“ F bO L defines an FG-module, and

¨ reduction modulo p of L, that is L :“ L{pL – k bO L defines a kG-module.

We note that, when seen as an O-module, an OG-module L may have torsion, which is lost onpassage to F . In order to avoid this issue, we usually only work with OG-lattices. In this way, weobtain functors

FG-mod OG-lat kG-mod

between the corresponding categories of finitely generated OG-lattices and finitely generated FG-,kG-modules.

A natural question to ask is: which FG-modules, respectively kG-modules, come from OG-lattices? Inthe case of FG-modules we have the following answer.

Proposition-Definition 31.6Let O be a complete discrete valuation ring and let F :“ FracpOq be the fraction field of O. Then,for any finitely generated FG-module V there exists an OG-lattice L which has an O-basis whichis also an F-basis. In this situation V – LF and we call L an O-form of V .

Proof : Choose an F-basis tv1, . . . , vnu of V and set L :“ OGv1 ` ¨ ¨ ¨ ÒGvn Ď V .Exercise: verify that L is as required.

On the other hand, the question has a negative answer for kG-modules.

Definition 31.7 (liftable kG-module)Let O be a commutative local ring with unique maximal ideal p :“ JpOq and residue field k :“ O{p.A kG-module M is called liftable if there exists an OG-lattice pM whose reduction modulo p of Mis isomorphic to M , that is

pM{p pM – M .

(Alternatively, it is also said that M is liftable to an OG-lattice, or liftable to O, or liftable tocharacteristic zero.)

Even though every OG-lattice can be reduced modulo p to produce a kG-module, not every kG-moduleis liftable to an OG-lattice.Being liftable for a kG-module is a rather rare property. However, the next section brings us someanswers towards classes of kG-modules which are liftable.


32 Lifting idempotentsThroughout this section, we assume that O is a complete discrete valuation ring with unique maximalideal p :“ JpOq and residue field k :“ O{p.

Moreover, we let A denote a finitely generated O-algebra of finite O-rank. Observe that since A is afinitely generated O-module, so is any simple A-module V and it follows from Nakayama’s lemma thatpV ‰ V , so pV “ 0. If m is a maximal left ideal of A, then A{m is a simple A-module and thereforem Ě pA. This proves that

pA Ď JpAq .

It follows that JpAq is the inverse image in A under the quotient morphism of the Jacobson radical JpAqof the finite dimensional k-algebra A :“ A{pA (the reduction modulo p of A). Consequently,

A{JpAq – A{JpAq .

Recall that an idempotent e of a ring R is called primitive if e ‰ 0 and whenever e “ f ` g where fand g are orthogonal idempotents, then either f “ 0 or g “ 0.

Exercise 32.1Let A be a ring and let e P A be an idempotent element. Prove that:

(a) JpeAeq “ eJpAqe;

(b) If A is a finite-dimensional k-algebra over a field k , then e is primitive if and only if eAe isa local ring. In that case eJpAqe is the unique maximal ideal of eAe.

This leads us to the following crucial result for representation theory of finite groups on the lifting ofidempotents.

Theorem 32.2 (Lifting theorem of idempotents (partial version))

Let A be a finitely generated O-algebra of finite O-rank. Set A :“ A{JpAq and for a P A writea :“ a` JpAq. The following assertions hold.

(a) If a P Aˆ

, then a P Aˆ. Thus there is a s.e.s. of groups

1 1` JpAq Aˆ Aˆ

1 .

(b) For any idempotent x P A, there exists an idempotent e P A such that e “ x .

(c) Two idempotents e, f P A are conjugate in A if and only if e and f are conjugate in A. Moreprecisely if e “ u f u´1, then u lifts to an invertible element u P Aˆ such that e “ ufu´1. Inparticular, if e “ f , then there exists u P p1` JpAqq Ď Aˆ such that e “ ufu´1.

(d) An idempotent e P A is primitive in A if and only if e is primitive in A.


(e) The quotient morphism AÑ A induces a bijection between the set PpAq of conjugacy classesof primitive idempotents of A and the set PpAq of conjugacy classes of primitive idempotentsof A.

Proof : (a) If a P A is not invertible, then Aa ‰ A. Then a P m for some maximal left ideal m of A by Zorn’slemma. Since m Ě JpAq, its image is a maximal left ideal of A and we have a P m. Thus a is notinvertible. The claim about the s.e.s. follows immediately.

(b) Let a1 P A such that a1 “ x and let b1 :“ a21 ´ a1. Define by induction two sequences of elements

of A :an :“ an´1 ` bn´1 ´ 2an´1bn´1 and bn :“ a2

n ´ an @n ě 2.

We now prove by induction that a2n ” an pmod JpAqnq, or in other words that bn P JpAqn. This is true

for n “ 1, and assuming that this holds for n, we have b2n P JpAqn`1 (because pJpAqnq2 Ď JpAqn`1),

and since a2n “ an ` bn we obtain

a2n`1 ” a2

n ` 2anbn ´ 4a2nbn pmod JpAqn`1q

“ an ` bn ` 2anbn ´ 4pan ` bnqbn” an ` bn ´ 2anbn “ an`1 pmod JpAqn`1q .

It follows that pbnqně1 converges to 0 and that panqně1 is a Cauchy sequence in the JpAq-adictopology. Since A is complete in the JpAq-adic topology (see Remark 30.7), panqně1 converges tosome element e P A. Clearly, e2 ´ e “ limnÑ8 bn “ 0, so e is an idempotent of A. Moreover

e “ a1 “ x ,

as required.

(c) It is clear that e and f are conjugate in A if e and f are conjugate in A (e “ ufu´1 ñ e “ u f u´1).Conversely, assume that there exists u P A

ˆ

such that e “ u f u´1. Then, by (a), u P Aˆ and so,replacing f by ufu´1 we can assume e “ f . Now let v :“ 1A ´ e´ f ` 2ef . Then, by (a), v P Aˆbecause

v “ 1A ´ e´ f ` 2ef “ 1A ´ e´ f ` 2ef “ 1A .

Moreover, ev “ ef “ vf and it follows that e “ evv´1 “ vfv´1, as required.(d) The idempotent e is primitive in A if and only if e and 0 are the only idempotents of eAe. Since

by the previous exercise JpeAeq “ eJpAqe “ JpAq X eAe, we have

eAe{JpeAeq “ eAe “ eAe .

Now, if f is a non-trivial idempotent of eAe, then by (b) applied to the algebra eAe, the idempotentf lifts to an idempotent f P eAe. This proves that e is primitive if e is primitive. Conversely if e isnot primitive, there exists a non-trivial idempotent f P eAe. Then f is not conjugate (that is, notequal) to 0 nor to the unity element e. Thus, it follows from (c) that f is a non-trivial idempotentof eAe, as required.

(e) This follows immediately from (b), (c) and (d).

A first possible application of Theorem 32.2 is a generalisation of this theorem providing a lifting ofidempotents from a quotient A{b for an arbitrary ideal b P A. (See [Thé95, (3.2) Theorem].) We statehere the particular case of interest to us for b “ pA.


Theorem 32.3 (Lifting theorem of idempotents for reduction modulo p)

Let A be a finitely generated O-algebra of finite O-rank. Set A :“ A{pA and for a P A writea :“ a` pA. The following assertions hold.

(a) For every idempotent x P A, there exists an idempotent e P A such that e “ x .

(b) Aˆ “ ta P A | a P Aû.

(c) If e1, e2 P A are idempotents such that e1 “ e2 then there is a unit u P Aˆ such thate1 “ ue2u´1 .

(d) The quotient morphism A� A induces a bijection between the central idempotents of A andthe central idempotents of A.

Proof : Exercise! (Either use Theorem 32.2 or imitate the proof of Theorem 32.2.)

The lifting of idempotents allows us, in particular, to prove that projective indecomposable kG-modulesare liftable to projective indecomposable OG-lattices.

Lemma 32.4Let A be a finitely generated algebra over a commutative ring R . If P is a projective indecomposableA-module, then there exists an idempotent e P A such that P – Ae.

Proof : Since P is projective, P | pA˝qn for some n P Zě1. As P is indecomposable, it follows from theKrull-Schmidt theorem that P | A˝, so A˝ “ P ‘Q for some A-module Q. Thus, we can write 1A “ e` fwith e P P and f P Q. Then

fe “ p1´ eqe “ e´ e2 “ ef

and this is an element of AeX Af Ď Q X P “ t0u. Therefore e2 “ e and ef “ fe “ 0. Finally, since

A “ A ¨ 1A “ Ape` fq “ Ae` Af , Ae Ď P and Af Ď Q ,

it follows that Ae “ P .

Corollary 32.5

Let A be a finitely generated O-algebra of finite O-rank and set A :“ A{pA. Let P be a projective(indecomposable) A-module. Then there exists a projective (indecomposable) A-module pP such thatP – pP{ppP .

Proof : Let P “ P1 ‘ ¨ ¨ ¨ ‘ Pr (r P Zě1) be a decomposition of P as a direct sum of indecomposable A-submodules. Then, by Lemma 32.4, there exist idempotents f1, . . . , fr P A such that Pi – Afi for each1 ď i ď r, and so

P – Af1 ‘ ¨ ¨ ¨ ‘ Afr .

Now, by Theorem 32.3(a) there exists idempotents e1, . . . , er P A such that ei “ fi for each 1 ď i ď r.Then pP :“ Ae1 ‘ ¨ ¨ ¨ ‘ Aer is a projective A-module (see Example 10) and pP{ppP – P .Moreover, pP is indecomposable if P is. Indeed,

pP “ pP1 ‘ pP2 decomposable ñ pP{ppP – k bO ppP1 ‘ pP2q – pk bO pP1q ‘ pk bO pP2q decomposable .


Corollary 32.6Any (projective) indecomposable kG-module is liftable to a (projective) indecomposable OG-lattice.

Proof : This follows immediately from Corollary 32.5 with A :“ OG since then

A “ OG{pOG – k bO OG – pk bO OqG – kG .

Exercise 32.7Let A be a finitely generated O-algebra of finite O-rank. Let M be an A-module and let e P A bean idempotent. Prove that, as EndApMq-modules,

HomApAe,Mq – eM .

33 Brauer ReciprocityThroughout this section, we let pF,O, kq be a splitting p-modular system for G and write p :“ JpOq.Reduction modulo p :“ JpOq provides us with a method for going from characteristic zero to character-istic p ą 0. Moreover, if V is an FG-module, then we can choose an O-form L of V and then considerthe reduction modulo p of L, i.e. L “ L{p. However, the choice of the O-form is not unique. As aconsequence, if L1 and L2 are two O-forms of V , i.e. pL1q

F – V – pL2qF , then it may happen that

L1 fl L2. The following result shows that, however, this does not affect the composition factors, up toisomorphism and multiplicity.

Proposition 33.1Let S1, . . . , St (t P Zě1) be a complete set of representatives of the isomorphism classes of simplekG-modules. If V is an FG-module and L is an O-form of V , then for each 1 ď j ď t the multiplicityof Sj as a composition factor of L “ L{pL does not depend on the choice of the O-form L.

Proof : Fix j P t1, . . . , tu. Since k is a splitting field for G, EndkGpSjq – k . Thus, by Proposition 22.1(b),the multiplicity of Sj as a composition factor of L is

dimk HomkGpPSj , Lq{ dimk EndkGpSjq “ dimk HomkGpPSj , Lq .

On the other hand, by Lemma 32.4 and Theorem 32.3, there exists an idempotent ej P OG such that

PSj – kGej .

Hence, Exercise 32.7 yields

HomkGpPSj , Lq – HomkGpkGej , Lq – ejL – ejL .

Then, Exercise 30.2(a)(ii) and Proposition-Definition 31.6 yield

dimk HomkGpPSj , Lq “ dimkpejLq “ rkOpejLq “ dimF pejV q .

As a consequence, for any 1 ď j ď t, the number of composition factors of L isomorphic to Sj is equal todimF pejV q, and is therefore independent of the choice of the O-form L.


Theorem 33.2 (Brauer Reciprocity)Let V1, . . . , Vl (l P Zě1) be a complete set of representatives of the isomorphism classes of simpleFG-modules, and let S1, . . . , St (t P Zě1) be a complete set of representatives of the isomorphismclasses of simple kG-modules. Let e1, . . . , et P OG be idempotents such that kGej is a projectivecover of Sj for each 1 ď j ď t. For every 1 ď i ď l and 1 ď j ď t define dij to be the multiplicityof Sj as a composition factor of the reduction modulo p of an O-form of Vi. Then

FGej –l

à

i“1dijVi.

Proof : Since FG is a semisimple F-algebra the set tVi | 1 ď i ď lu is a complete set of representativesof the isomorphism classes of the PIMs of FG (by Theorem 21.2(b)). Moreover, as F is a splitting fieldfor G, by Theorem 8.2, each Vi (1 ď i ď l) appears precisely dimF Vi times in the decomposition of theregular module FG. Hence, for any 1 ď j ď t, there exist non-negative integers d1ij such that

FGej “l

à

i“1d1ijVi ,

where d1ij “ dimF HomFGpFGej , Viq. Thus, it remains to prove that d1ij “ dij for every 1 ď i ď l and1 ď j ď t. So choose an O-form Li of Vi (1 ď i ď). As in the previous proof, applying Exercise 32.7 andProposition 31.6 yields

d1ij “ dimF HomFGpFGej , Viq “ dimF pejViq “ rkOpejLiq “ dimk ejLi “ dimk HomkGpkGej , Liq “ dij .

Definition 33.3 (Decomposition matrix)The positive integers dij (1 ď i ď l, 1 ď j ď t) defined in Theorem 33.2 are called the p-decomposition numbers of G and the matrix

DecppGq :“ pdijq1ďiďl1ďjďt

is called the p-decomposition matrix of G (or simply the decomposition matrix) of the finite group G.

Exercise 33.4Let C be the Cartan matrix of kG and D :“ DecppGq be the decomposition matrix of G. Prove that

C “ DtrD

and deduce again that C is symmetric.

Chapter 9. Brauer Characters

Recall that if F is a field of characteristic zero, then FG-modules are isomorphic if and only if theircharacters are equal. Also, the character of a FG-module provides complete information about itscomposition factors, including multiplicities, provided that the irreducible characters are known. Allthis does not hold for fields k of characteristic p ą 0. For instance, if W is a k-vector space on whichG acts trivially and dimkpW q “ ap ` 1 for some nonnegative integer a, then the k-character of W isthe trivial character. This implies that a k-character can only give information about multiplicities ofcomposition factors modulo p. In view of these issues, the aim of this chapter is to define a slightlydifferent kind of character theory for modular representations of finite groups and to establish linkswith ordinary character theory.

Notation: Throughout, G denotes a finite group and p a prime number. We let pF,O, kq denote ap-modular system and we assume F contains all exppGq-th roots of unity, so pF,O, kq is a splitting p-modular system for G and all its subgroups (see Theorem 31.2). We write p :“ JpOq. For Λ P tF,O, kuall ΛG-modules considered are assumed to be free of finite rank over Λ. If Λ P tF, ku and U is aΛG-module, then we write ρU : G ÝÑ GLpUq for the associated Λ-representation.

For background results in ordinary character theory I refer to my Skript Character Theory of FiniteGroups from the SS 2020.

References:



[Isa94] I. M. Isaacs. Character theory of finite groups. Dover Publications, Inc., New York, 1994.[Las20] C. Lassueur. Character theory of finite groups. Lecture Notes SS 2020, TU Kaiserslautern,

2020.[NT89] H. Nagao and Y. Tsushima. Representations of finite groups. Academic Press, Inc., Boston,

MA, 1989.[Nav98] G. Navarro. Characters and blocks of finite groups. Cambridge University Press, Cambridge,

1998.[Web16] P. Webb. A course in finite group representation theory. Vol. 161. Cambridge Studies in

Advanced Mathematics. Cambridge University Press, Cambridge, 2016.

93


Before we start, we recall the following useful terminology from finite group theory.

DefinitionLet G be a finite group and p be a prime number.

(a) If |G| “ pam with a,m P Zě0 and pp,mq “ 1, then |G|p :“ pa is the p-part of the order |G|of G and |G|p1 :“ m is the p1-part of the order of G.

(b) An element g P G is called a p-regular element (or a p1-element) if p - opgq and we write

Gp1 :“ tg P G | p - opgqu

for the set of all p-regular elements of G. (Warning: in general this is not a subgroup!)

(c) An element g P G is called a p-singular element if p | opgq and it is called a p-element ifopgq is a power of p.

RemarkLet G be a finite group and p be a prime number. Let g P G and write opgq “ pnm with n,m P Zě0such that pp,mq “ 1. Then, letting a, b P Z be such that 1 “ apn ` bm, we may set

gp :“ gbm and gp1 :“ gapn

It is immediate thatpgpqp

n“ 1, pgp1qm “ 1, and g “ gpgp1 .

Thus, gp is a p-element and is called the p-part of g, and gp1 is p-regular and is called the p1-partof g.

34 Brauer charactersSince we assume that the given p-modular system pF,O, kq is such that F contains all exppGq-th rootsof unity, both F and k contain a primitive a-th root of unity, where a is the l.c.m. of the orders of thep-regular elements. To start with we examine the relationship between the roots of unity in F and ink . Set

µF :“ ta-th roots of 1 in Fu ;

µk :“ ta-th roots of 1 in ku .

Then µF Ď O and, as both µF and µk are finite groups, it follows from Corollary 30.8 that the quotientmorphism O� O{p restricted to µF induces a group isomorphism

µF–ÝÑ µk .

We write the underlying bijection as pξ ÞÑ ξ , so that if ξ is an a-th root of unity in k then pξ is theunique a-th root of unity in O which maps onto it.


Lemma 34.1 (Diagonalisation lemma)Let ρ : G ÝÑ GLpUq be a k-representation of G. Then, for every p-regular element g P Gp1 , thek-linear map ρpgq is diagonalisable and the eigenvalues of ρpgq are opgq-th roots of unity and liein µk . In other words, there exists an ordered k-basis B of U with respect to which

`

ρpgq˘

B “

»

—

—

—

—

–

ξ1 0 00 ξ2

00 0 ξn

fi

ffi

ffi

ffi

ffi

fl

,

where n :“ dimkpUq and each ξi (1 ď i ď n) is an opgq-th root of unity in k .

Proof : Let g P Gp1 . It is enough to consider the restriction of ρ to the cyclic subgroup xgy. Since p - |xgy|,kxgy is semisimple by Maschke’s Theorem. Moreover, as k is a splitting field for xgy, it follows fromCorollary 12.3 that all irreducible k-representations of xgy have degree 1. Hence ρ|xgy can be decomposedas the direct sum of degree 1 subrepresentations. As a consequence ρpgq “ ρ|xgypgq is diagonalisableand there exists a k-basis B of U satisfying the statement of the lemma. It follows immediately thatthe eigenvalues are opgq-th roots of unity because ρUpgopgqq “ ρUp1Gq “ IdU . They all lie in µk , beingopgq-th roots of unity, hence a-th roots of unity.

This leads to the following definition.

Definition 34.2 (Brauer characters)Let U be a kG-module of dimension n P Zě1 and let ρU : G Ñ GLpUq be the associated k-representation. The p-Brauer character or simply the Brauer character of G afforded by U (resp.of ρU ) is the F-valued function

φU : Gp1 Ñ O Ď F

g ÞÑ pξ1 ` ¨ ¨ ¨ ` pξn ,

where ξ1, . . . , ξn P µk are the eigenvalues of ρUpgq. The integer n is also called the degree of φU .Moreover, φU is called irreducible if U is simple (resp. if ρU is irreducible), and it is called linearif n “ 1. We denote by IBrppGq the set of all irreducible Brauer characters of G and we write 1Gp1for the Brauer character of the trivial kG-module.

In the sequel, we want to prove that Brauer characters of kG-modules have properties similar to C-characters.

Remark 34.3

(a) Warning: φpgq P O Ď F even though ρUpgq is defined over the field k of characteristic p ą 0.

(b) Often the values of Brauer characters are considered as complex numbers, i.e. sums of complexroots of unity. Of course, in that case then φUpgq depends on the choice of embedding ofµF into C. However, for a fixed embedding, φUpgq is uniquely determined up to similarityof ρUpgq.


Immediate properties of Brauer characters are as follows.

Proposition 34.4Let U ‰ 0 be a kG-module with Brauer character φU . Then:

(a) φUp1q “ dimkpUq ;

(b) φU is a class function on Gp1 ;

(c) φUpg´1q “ φU˚pgq @ g P Gp1 .

Proof : Let n :“ dimkpUq and let ρU be the k-representation associated to U .

(a) Clearly ρUp1Gq is the identity map on U and has n eigenvalues all equal to 1k . Since p1k “ 1F thesum of the lifts is n “ dimkpUq.

(b) If g, x P Gp1 , then ρUpxgx´1q “ ρUpxqρUpgqρUpxq´1, so ρUpgq and ρUpxgx´1q are similar andtherefore have the same eigenvalues. Thus, by definition of φU we have φUpgq “ φUpxgx´1q.[Note that in this proof we could take x P G and it would not change the result!]

(c) Let g P Gp1 and let ξ1, . . . , ξn P µk be the eigenvalues of ρUpgq. As ρUpgq is diagonalisable byLemma 34.1, it follows immediately that the eigenvalues of ρUpg´1q “ ρUpgq´1 are ξ´1

1 , . . . , ξ´1n ,

as are the eigenvalues of ρU˚pgq “ ρUpg´1qtr . Since µF ÝÑ µk , pξ ÞÑ ξ is a group isomorphism,we have yξ´1

i “ pξi´1

for each 1 ď i ď n, and the claim follows.

Exercise 34.5Let U,V ,W be non-zero kG-modules. Prove the following assertions.

(a) If 0 U V W 0 is a s.e.s. of kG-modules, then

φV “ φU ` φW .

(b) If the composition factors of U are S1, . . . , Sm (m P Zě1) with multiplicities n1, . . . , nm re-spectively, then

φU “ n1φS1 ` . . .` nmφSm .

In particular, if two kG-modules have isomorphic composition factors, counting multiplicities,then they have the same Brauer character.

(c) φU‘V “ φU ` φV and φUbkV “ φU ¨ φV .

The link between the Brauer characters and the usual notion of a character, i.e. trace functions, isdescribed below and explains why trace functions are not good enough.

Definition 34.6 (character )Let Λ P tF, ku. Let U ‰ 0 be a ΛG-module and let ρU : G ÝÑ GLpUq be the associated Λ-representation, then the trace function

Trp´, Uq : G ÝÑ Λg ÞÑ Trpg,Uq :“ Tr

`

ρUpgq˘


is called the Λ-character of U .For Λ “ F , we also write χU instead of Trp´, Uq and we write IrrF pGq for the set of all irreducibleF-characters of G.

(Recall from Character Theory of Finite Groups that an F-character χU is called irreducible if U is asimple FG-module.)

Lemma 34.7Let U ‰ 0 be a kG-module and let g P G. Then:

(a) Trpg,Uq “ Trpgp1 , Uq , and

(b) Trpg,Uq “ φUpgp1q ` p in the quotient k “ O{p.

Proof : Let ρU be the k-representation associated to U .

(a) Clearly ρUpgq “ ρUpgpqρUpgp1q “ ρUpgp1qρUpgpq. Therefore, we know from linear algebra that theeigenvalues of ρUpgq are pairwise product of suitably ordered eigenvalues of ρUpgp1q and ρUpgpq.Now, as gp is a p-element, a p-power of ρUpgpq is the identity and so all the eigenvalues of ρUpgpqare equal to one because charpkq “ p. Hence ρUpgq and ρUpgp1q have the same eigenvalues(counted with multiplicities) and the claim follows.

(b) Let ξ1, . . . , ξn be the eigenvalues of ρUpgp1q. Then, by definition, φUpgq “ pξ1 ` ¨ ¨ ¨ ` pξn, where, foreach 1 ď i ď n, pξi is such that pξi ` p “ ξi in the quotient k “ O{p. Therefore, it follows from (a)and the diagonalisation lemma that

Trpg,Uq “ Trpgp1 , Uq “ ξ1`¨ ¨ ¨`ξn “ p pξ1`pq`¨ ¨ ¨`p pξn`pq “ p pξ1` . . .` pξnq`p “ φUpgp1q`p .

Notation 34.8We let ClF pGq :“ tf : G ÝÑ F | f class functionu denote the F-vector space of F-valued classfunctions on G and we let ClF pGp1q :“ tf : Gp1 ÝÑ F | f class functionu denote the F-vector spaceof F-valued class functions on Gp1 .

Proposition 34.9A class function φ P ClpGp1q is a Brauer character if and only if it is a non-zero non-negativeZ-linear combination of elements of IBrppGq.

Proof : The necessary condition is clear by Exercise 34.5(b). Conversely, let n1, . . . , nm P Zě0 and letφS1

, . . . , φSm P IBrppGq pm P Zě1) be the Brauer characters afforded by simple kG-modules S1, . . . , Sm.Then, by Exercise 34.5(c), the class function φ :“ n1φS1

` . . .` nmφSm is the Brauer character affordedby the kG-module

Sn11 ‘ . . .‘ Snmm .

Theorem 34.10The set IBrppGq of irreducible Brauer characters of G is F-linearly independent and hence

| IBrppGq| ď dimF ClF pGp1q “ number of conjugacy classes of p-regular elements in G .

In particular, IBrppGq “ t1Gp1 u provided G is a finite p-group.


Without proof.The second claim is obvious, because the indicator functions on the conjugacy classes of p-regularelements form an F-basis. The third claim is clear since if G is a p-group, then Gp1 “ t1Gu. Thus| IBrppGq| “ 1 by the first part, namely IBrppGq “ t1Gp1 u.

35 Back to decomposition matrices of finite groupsWe now want to investigate the connections between representations of G over F (or C) and represen-tations of G over k through the connections between their F-characters and Brauer characters.

Lemma 35.1

Let V be an FG-module with F-character χV . Let L be an O-form of V and let L “ L{pL be thereduction modulo p of L. Then,

χV |Gp1 “ φL ,

and is often called the reduction modulo p of χV .

Proof : Let g P Gp1 and write RL : G ÝÑ GLpOq for the O-representation associated to L. Since g isp-regular, the eigenvalues x1, . . . , xn of RLpgq belong to µF Ď O and x1, . . . , xn P k are the eigenvaluesof ρLpgq. Since by definition φL is the sum of the lifts of the latter quantities, we have

φLpgq “ x1 ` . . .` xn “ Trpg, LF q “ Trpg, V q “ χV pgq

since L is an O-form of V .

Proposition 35.2

(a) The set IBrppGq is an F-basis of ClF pGp1q.

(b) Given an irreducible F-character χ P IrrF pGq, there exist non-negative integers dχφ such that

χ |Gp1 “ÿ

φPIBrppGqdχφφ.

(c) | IBrppGq| “ number of p-regular conjugacy classes of G .

Proof : By Theorem 34.10 it only remains to prove that IBrppGq spans ClF pGp1q. So let µ P ClF pGp1q bean arbitrary F-valued class function on Gp1 . Then µ can be extended to µ# P ClF pGq, e.g. by settingµ#pgq “ 0 for every g P GzGp1 . Since IrrF pGq is an F-basis for ClF pGq, we can write

µ# “ÿ

χPIrrF pGqaχ ¨ χ

with aχ P F @ χ P IrrF pGq. Thus

µ “ µ#|Gp1 “ÿ

χPIrrF pGqaχ ¨ χ |Gp1 .


where by Lemma 35.1 each χ |Gp1 is a Brauer character of G, and hence by Proposition 34.9, each χ |Gp1is a non-negative integer linear combination of irreducible Brauer characters. Therefore µ is an F-linearcombination of irreducible Brauer characters over F . It follows that IBrppGq is a generating set forClF pGp1q. This proves (a). Parts (b) and (c) are immediate consequences of (a) and its proof.

Exercise 35.3Prove that two kG-modules afford the same Brauer character if and only if they have isomorphiccomposition factors (including multiplicities).

Remark 35.4If we translate part (b) of Corollary 35.2 from irreducible characters and Brauer characters to FG-modules and kG-modules, we obtain that the integers dχφ are the same as the p-decompositionnumbers of G as defined through Brauer’s reciprocity theorem, i.e.

D :“ DecppGq “ pdχφqχPIrrF pGqφPIBrppGq

.

The Cartan matrix of G is then

C “ DtrD “ pcφµqφ,µPIBrppGq

(see Exercise 33.4) and for every φ, µ P IBrppGq we have

cφµ “ÿ

χPIrrF pGqdχφdχµ.

Corollary 35.5

(a) The decomposition matrix DecppGq of G has full rank, namely | IBrppGq|.

(b) The Cartan matrix of G is a symmetric positive definite matrix with non-negative integerentries.

Proof : It follows from Proposition 35.2 that tχ |Gp1 | χ P IrrF pGqu spans ClF pGp1q over F . There is thereforea subset B Ď tχ |Gp1 | χ P IrrF pGqu which forms an F-basis for ClF pGp1q. Now by Proposition 35.2, thecolumns of the matrix pdχφqχPB,φPIBrppGq are F-linearly independent, hence DecppGq has full rank. Thisproves (a) and (b) follows immediately.

Recall now that projective kG-modules are liftable by Corollary 32.5. This enables us to associate anF-character of G to each PIM of kG.

Definition 35.6Let φ P IBrppGq be an irreducible Brauer character afforded by a simple kG-module S. Let PS bethe projective cover of S and let pPS denote a lift of PS to O. Then, the F-character of ppPSqF isdenoted by Φφ and is called the projective indecomposable character associated to S or φ.


Corollary 35.7Let φ P IBrppGq. Then:

(a) Φφ “ř

χPIrrF pGq dχφχ ; and

(b) Φφ|Gp1“

ř

µPIBrppGq cφµµ.

Proof : (a) This follows from Brauer reciprocity.(b) This follows from part (a) because

Φφ|Gp1 “ÿ

χPIrrF pGqdχφχ |Gp1 “

ÿ

χPIrrF pGqdχφ

ÿ

µPIBrppGqdχµµ “

ÿ

µPIBrppGqcφµµ .

Theorem 35.8Assume p - |G|, then the following assertions hold:

(a) If V is a simple FG-module and L is an O-form for V , then its reduction modulo p is a simplekG-module and the map

IrrF pGq ÝÑ IBrppGqχV ÞÑ χV |Gp1 “ χV

is a bijection.

(b) Both the Cartan matrix of G and the decomposition matrix DecppGq of G are the identitymatrix, provided the rows and the columns are ordered in the same way.

Proof : Since p - |G|, clearly Gp1 “ G. Now, by Maschke’s theorem, kG is semisimple, so the simple kG-modules are precisely the PIMs of kG. Thus, if S is a simple kG-module affording the Brauer character φ,then by definition of the Cartan integers (Def. 22.2) and Remark 35.4 we have

cφµ “ δφµ @ µ P IBrpG .

Therefore,1 “ cφφ “

ÿ

χPIrrF pGqd2χφ

and it follows that there is a unique χ0 P IrrF pGq such that dχ0φ ‰ 0; in fact we must have dχ0φ “ 1.Now, it follows from Corollary 35.7 that

Φφ “ χ0 and χ0|Gp1 “ φ ,

so the first claim of (a) holds and the given map is well-defined and surjective. It follows immediatelythat f is bijective since by Proposition 35.2(c) we have

| IBrppGq| “ #G-conjugacy classes in G “ | IrrF pGq| ă 8 .

This proves (a) and (b).


Finally, we would like to obtain orthogonality relations for Brauer characters, which generalise therow orthogonality relations for ordinary irreducible characters. However, unlike the case of ordinarycharacters, there are now two significant tables that we can construct: the table of values of Brauercharacters of simple modules, and the table of values of Brauer characters of indecomposable projectivemodules.

Definition 35.9 (Brauer character table)Set l :“ |Gp1 | and let g1, . . . , gl be a complete set of representatives of the p-regular conjugacyclasses of G.

(a) The Brauer character table of a finite group G is the matrix´

φpgjq¯

φPIBrppGq1ďjďl

P MlpF q .

(b) The Brauer projective table of a finite group G at p is the matrix´

Φφpgjq¯

φPIBrppGq1ďjďl

P MlpF q .

Definition 35.10Given F-valued class functions f1, f2 defined on a subset of G containing Gp1 , define

xf1, f2yp1 :“ 1|G|

ÿ

gPGp1f1pgqf2pg´1q .

Note. It is straightforward that x , yp1 is F-bilinear on ClF pGp1q.

With these tools we obtain a replacement for the row orthogonality relations for ordinary irreduciblecharacters. It says that the rows of the Brauer character table and of the p-projective Brauer table areorthogonal to each other.

Theorem 35.11

(a) All projective indecomposable characters Φφ (φ P IBrppGq) vanish on p-singular elements,and the set tΦφ | φ P IBrppGqu is an F-basis of the subspace

Cl˝F pGq :“ tf P ClF pGq | fpgq “ 0 @g P GzGp1u

of ClF pGq.

(b) For every φ, ψ P IBrppGq we have xφ,Φψyp1 “ δφψ “ xΦφ, ψyp1 .

Proof : Let g1, . . . , gr be a complete set of representatives of the conjugacy classes of G such that g1, . . . , glare p-singular and gl`1, . . . , gr are p-regular. If l ă j ď r, then by Proposition 35.2(b),

χpg´1j q “ χ |Gp1 pg

´1j q “

ÿ

φPIBrppGqdχφφpg´1

j q

and for each 1 ď i ď r the 2nd Orthogonality Relations for the irreducible F-characters (see [Las20,


Thm. 12.2]) yield

p˚q δij |CGpgiq| “ÿ

χPIrrF pGqχpgiqχpg´1

j q “ÿ

χPIrrF pGqχpgiq

´

ÿ

φPIBrppGqdχφφpg´1

j q¯

“ÿ

φPIBrppGq

´

ÿ

χPIrrF pGqdχφχpgiq

¯

φpg´1j q

“ÿ

φPIBrppGqΦφpgiqφpg´1

j q

where the last equality holds by Corollary 35.7(a). Then for each 1 ď i ď l, we haveř

φPIBrppGq Φφpgiqφ “0in ClF pGp1q and we obtain that Φφpgiq “ 0 for each φ P IBrppGq since IBrppGq is F-linearly independent.Thus the first claim of (a) is established.Now, (˚) says that the square matrix

ˆ

Φφpgiq|CGpgiq|

˙tr

φPIBrppGql`1ďiďr

is a left inverse for the square matrixˆ

φpg´1i q

˙

φPIBrppGql`1ďiďr

.

Therefore, it is also a right inverse, and multiplying both matrices the other way around, we obtain(applying the Orbit-Stabiliser Theorem) that

δφψ “rÿ

i“l`1

1|CGpgiq|

φpg´1i qΦψpgiq “

1|G|

ÿ

gPGp1φpg´1qΦψpgq

“1|G|

ÿ

gPGp1φpgqΦψpg´1q

“ xφ,Φψyp1

for all φ, ψ P IBrppGq. Similarly xΦφ, ψyp1 “ δφψ for all φ, ψ P IBrppGq. This means in effect thattΦφ|Gp1 | φ P IBrppGqu and IBrppGq are dual bases of ClF pGp1q with respect to the F-bilinear form x, yp1 .Thus, we have proved (a) and (b).

Remark 35.12The proof of the theorem tells us that writing Φ for the Brauer projective table, Π for the Brauercharacter table and setting B :“ diagp|CGpgl`1q|, . . . , |CGpgrq|q, then the orthogonality relationscan be written as ΦtrB´1Π “ I .

Exercise 35.13Let H be a p1-subgroup of a finite group G. Prove that the character Φk is a constituent of thetrivial F-character of H induced to G.

Exercise 35.14Let φ, λ P IBrppGq and assume that λ is linear. Prove that λφ P IBrppGq and λΦφ “ Φλφ .

Exercise 35.15Let G be a finite group and let ρreg denote the regular F-character of G. Prove that:

ρreg “ÿ

φPIBrppGqφp1qΦφ and pρregq|Gp1 “

ÿ

φPIBrppGqΦφp1qφ .


Exercise 35.16Deduce from Theorem 35.11 that:

(a) the inverse of the Cartan matrix of kG is C´1 “ pxφ, ψyp1qφ,ψPIBrppGq; and

(b) |G|p | Φφp1q for every φ P IBrppGq.

Exercise 35.17

(a) Let U be a kG-module and let P be a PIM of kG. Prove that

dimk HomkGpP,Uq “1|G|

ÿ

gPGp1φPpg´1qφUpgq

(b) Prove that the binary operation x , yp1 is an inner product on ClF pGp1q.

Exercise 35.18Let G :“ A5, the alternating group on 5 letters. Calculate the Brauer character table, the Cartanmatrix and the decomposition matrix of G for p “ 3.[Hints. (1.) Use the ordinary character table of A5 and reduction modulo p. (2.) A simple group does not have anyirreducible Brauer character of degree 2.]

Exercise 35.19Deduce from Remark 35.12 that column orthogonality relations for the Brauer characters take theform ΠtrΦ “ B, i.e. given g, h P Gp1 we have

ÿ

φPIBrppGqφpgqΦφph´1q “

#

|CGpgq| if g and h are G-conjugate,0 otherwise.

Chapter 10. Blocks

We can break down the representation theory of finite groups into its smallest parts: the blocks of thegroup algebra. First we will define blocks for arbitrary rings and then specify to the situation of a finitegroup G and a splitting p-modular system pF,O, kq.

Notation: Throughout, G denotes a finite group, p a prime number. We let pF,O, kq denote a p-modularsystem and we assume F contains all exppGq-th roots of unity, so pF,O, kq is a splitting p-modularsystem for G and all its subgroups (see Theorem 31.2). We write p :“ JpOq. For Λ P tF,O, ku allΛG-modules considered are assumed to be free of finite rank over Λ.

References:


[Lin18] M. Linckelmann. The block theory of finite group algebras. Vol. I. Vol. 91. London Mathemat-ical Society Student Texts. Cambridge University Press, Cambridge, 2018.




36 The blocks of a ringThroughout this section we let A denote an arbitrary associative ring with an identity element (de-noted 1A).

Proposition 36.1

(a) There is a bijection between

(1) the set of decompositions

A “ A1 ‘ ¨ ¨ ¨ ‘ Ar pr P Zě1q

of A into a direct sum of pA, Aq-subbimodules (or equiv. of two-sided ideals of A), and

104


(2) the set of decompositions1A “ e1 ` ¨ ¨ ¨ ` er

of 1A into a sum of orthogonal idempotents of Z pAq,

in such a way that ei “ 1Ai and Ai “ Aei for every 1 ď i ď r.

(b) For each 1 ď i ď r, the direct summand Ai of A is indecomposable as an pA, Aq-bimodule ifand only if the corresponding central idempotent ei is primitive.

(c) The decomposition of A into indecomposable pA, Aq-subbimodules is uniquely determined.

Proof :(a) Assume A “ A1 ‘ ¨ ¨ ¨ ‘ Ar is a decomposition of A into pA, Aq-bimodules. Then for each 1 ď i ď r,

there exists ei P Ai such that1A “ e1 ` ¨ ¨ ¨ ` er .

Furthermore, for any element a P A,

a “ 1A ¨ a “ pe1 ` ¨ ¨ ¨ ` erqa “ e1a` ¨ ¨ ¨ ` era

anda “ a ¨ 1A “ ape1 ` ¨ ¨ ¨ ` erq “ ae1 ` ¨ ¨ ¨ ` aer .

So for each 1 ď j ď r, we have eja “ aej P Aj , and it follows in particular that e1, . . . , er P Z pAq.Also, if ai P Ai then eiai “ ai and ejai “ 0 if j ‰ i. Hence, ei “ 1Ai and e2

i “ ei so teiuri“1 is aset of orthogonal idempotents of Z pAq, and Ai “ Aei for 1 ď i ď r.Conversely, if teiuri“1 is a set of orthogonal idempotents of Z pAq such that 1A “

řri“1 ei, then

Ai :“ Aei is an pA, Aq-subbimodule of A for each 1 ď i ď r and A1 ‘ ¨ ¨ ¨ ‘ Ar is a direct sumdecomposition of A into pA, Aq-subbimodules.

(b) "ñ" Consider Ai “ Aei and suppose there exist orthogonal idempotents f , h P Z pAq such thatei “ f ` h. Then Ai “ Af ‘ Ah, where the sum is direct because if a P Af X Ah then a “ afand a “ ah, hence a “ afh “ a ¨ 0 “ 0. Thus, if ei is not primitive, then Ai is decomposableas an pA, Aq-bimodule.

"ð" Now suppose that Ai “ Aei “ L1 ‘ L2, where L1 and L2 are two non-zero pA, Aq-submodules.Then there exist f P L1zt0u and h P L2zt0u such that ei “ f ` h. Now, fh “ 0 sincefh P L1 X L2 “ t0u. As ei is the identity in Ai, we get

f “ eif “ pf ` hqf “ f2 ` hf “ f2 ` 0 “ f2

and similarly, h2 “ h, hence f and h are orthogonal idempotents, so ei is not primitive.(c) Suppose that A “ A1 ‘ ¨ ¨ ¨ ‘ Ar (r P Zrě1q is a decomposition of A into pA, Aq-subbimodules, and

suppose that L is an indecomposable direct summand of A from a different such decomposition of A.Every x P L can be written as

x “ a1 ` ¨ ¨ ¨ ` ar with ai P Ai for each 1 ď i ď r,

so L Q eix “ ai. Hence L “ pL X A1q ‘ ¨ ¨ ¨ ‘ pL X Arq and this is a decomposition of L. Since Lis indecomposable there exists 1 ď m ď r such that L “ L X Am. By the indecomposability of Am,this must be the whole of Am. Thus the decomposition of A into pA, Aq-subbimodules is uniquelydetermined.


Definition 36.2 (Block, block idempotent, block algebra, belonging to a block )

(a) The uniquely determined pA, Aq-bimodules Ai “ Aei (1 ď i ď r) given by Theorem 36.1(c) arecalled the blocks of A and the corresponding primitive central idempotents ei are called theassociated block idempotents.(The blocks are sometimes also called block algebras when A is an algebra.)

(b) We say that an (indecomposable) A-module M belongs to (or lies in) the block Ai “ Aei ifeiM “ M and ejM “ 0 for all j ‰ i.

Exercise 36.3Let A “ A1 ‘ ¨ ¨ ¨ ‘ Ar (r P Zě1) be the block decomposition of A and let M be an arbitrary A-module. Prove that M admits a unique direct sum decomposition M “ M1 ‘ ¨ ¨ ¨ ‘Mr where foreach 1 ď i ď r the summand Mi belongs to the block Ai of A . Deduce that every indecomposableA-module lies in a uniquely determined block of A.

Corollary 36.4Let A “ A1 ‘ ¨ ¨ ¨ ‘ Ar (r P Zě1) be the block decomposition of A and let 0 L M N 0be a s.e.s. of A-modules and A-module homomorphisms. Then, for each 1 ď i ď r:

M lies in the block Ai of A if and only L and N lie in Ai .

In particular, if an A-module M lies in a block Ai of A, then so do all of its submodules and all ofits factor modules.

Proof : Let ei P Z pAq be the primitive idempotent corresponding to Ai. Now, by Definition 36.2 an A-modulebelongs to the block Ai “ Aei if and only if external multiplication by ei is an A-isomorphism on thatmodule. Considering the commutative diagram

0 L M N 0

0 L M N 0

– ei¨p´q ei¨p´q ei¨p´q –

it follows from the five-Lemma that this property holds for M if and only if it holds for L and N .

37 p-Blocks of finite groupsWe now return to finite groups and observe the following.

Example 17 (Blocks of FG)Since FG is semisimple, the block decomposition of FG is given by the Artin-Wedderburn Theorem.In particular, the blocks are matrix algebras and can be labelled by the isomorphism classes ofsimple KG-modules.


Remark 37.1 (Blocks of OG and kG)The lifting of idempotents obtained in Theorem 32.3 tells us that the quotient morphismOG � rO{psG “ kG, x ÞÑ x :“ x ` p ÖG induces a bijection

tprimitive idempotents of Z pOGqu „ÐÑ tprimitive idempotents of Z pkGqu

e ÞÑ e

Thus a decomposition 1OG “ e1 ` ¨ ¨ ¨ ` er of the identity element of OG into a sum of primitivecentral idempotents corresponds to a decomposition 1kG “ e1 ` ¨ ¨ ¨ ` er of the identity element ofkG into a sum of primitive central idempotents of kG. Therefore, by Proposition 36.1, there is abijection between the blocks of OG and the blocks of kG.

Warning: The definition of a block of a finite group may vary from one book (resp. article) to anotherand different authors use different definitions of blocks of finite groups which can, from text to text, bealgebras, idempotents, sets of modules, sets of ordinary characters and/or Brauer characters. This maysound confusing at first sight, but in general, the context makes it clear which kind of objects is meant.

Definition 37.2 (Blocks of finite groups, principal block )Let Λ P tF,O, ku. Then:

(a) The blocks of ΛG are the uniquely determined indecomposable pΛG,ΛGq-bimodules definedby Theorem 36.1(c).

(b) We define a p-block of G to be the specification of a block of OG, understanding also thecorresponding block of kG, the corresponding idempotents of OG or of kG, or the moduleswhich belongs to these blocks. We write BlppGq for the set of all p-blocks of G, resp. BlppkGqfor the set of all blocks of kG.

Example 18 (p-block(s) of a p-group)If G is a p-group, then G a unique p-block. Indeed, we have already observed that in this case,the trivial module is the unique simple module and hence a unique PIM, namely kG itself. So kGis certainly also indecomposable as a pkG, kGq-bimodule.

Theorem 37.3Let K P tF, ku and let S fl T be simple KG-modules. Then the following assertions are equivalent:

(a) S and T belong to the same block of KG.

(b) There exist simple KG-modules S “ S1, . . . , Sm “ T pm P Zě2q such that Si and Si`1 are bothcomposition factors of the same projective indecomposable KG-module for each 1 ď i ď m´1.

Proof : It is clear that all (a) and (b) define equivalence relations on the set of simple KG-modules. Wehave to prove that it is the same relation.

(b)ñ(a): This follows by induction on m because by Corollary 36.4 all the composition factors of a PIM ofKG belong to the same block.

(a)ñ(b): Assume S belongs to the block B of KG. Decompose B “ P1 ‘ ¨ ¨ ¨ ‘ Ps ‘ Q1 ‘ ¨ ¨ ¨ ‘ Qt inPIMs where composition factors of the Pi’s are in relation (b) with S but none of the compositionfactors of the Qj ’s are in relation (b) with S. To prove: t “ 0. Clearly: then all simple modules T


belonging to B are in relation (b) with S by Corollary 36.4 since its projective cover must be oneof the Pi’s. Now, by construction, no composition factor of the Qj ’s can be a composition factor ofthe Pi’s, so HomBpPi, Qjq “ 0 “ HomBpQj , Piq for every 1 ď i ď s and every 1 ď j ď r. Thusboth P1‘ ¨ ¨ ¨ ‘Ps and Q1‘ ¨ ¨ ¨ ‘Qt are invariant under all B-endomorphisms, in particular underleft and right multiplication by elements of B, so they must themselves be blocks of KG. The onlypossibility is t “ 0.

The effect of this result is that the division of the simple kG-modules into blocks can be achieved ina purely combinatorial fashion, knowing the Cartan matrix of kG. The connection with a block matrixdecomposition of the Cartan matrix is probably the origin of the use of the term block in representationtheory.

Corollary 37.4On listing the simple kG-modules so that modules in each block occur together, the Cartan matrix ofkG has a block diagonal form, with one block matrix for each p-block of the group. Up to permutationof simple modules within p-blocks and permutation of the p-blocks, this is the unique decompositionof the Cartan matrix into block diagonal form with the maximum number of block matrices.

Proof : Given any matrix we may define an equivalence relation on the set of rows and columns of the matrixby requiring that a row be equivalent to a column if and only if the entry in that row and column isnon-zero, and extending this by transitivity to an equivalence relation. In the case of the Cartan matrixthe row indexed by a simple module S is in the same equivalence class as the column indexed by PS ,because S is a composition factor of PS . If we order the rows and columns of the Cartan matrix so thatthe rows and columns in each equivalence class come together, the matrix is in block diagonal form,with square blocks, and this is the unique such expression with the maximal number of blocks (up topermutation of the blocks and permutation of rows and columns within a block). It follows Theorem 37.3that the matrix blocks biject with the blocks of the group algebra.

38 Defect GroupsFrom now on we will only discuss the blocks of kG. (Analogous results hold for the correspondingblocks of OG.) To finish this chapter, for the sake of completeness, in this section and the next sectionwe give an overview of important results concerning p-blocks, mostly with sketches of proofs only.

Remark 38.1Observe that any pkG, kGq-bimodule B becomes a left krG ˆ Gs-module in a natural way via theG ˆ G-action

pG ˆ Gq ˆ B Ñ B`

pg, hq, b˘

ÞÑ gbh´1

extended by k-bilinearity to the whole of krG ˆ Gs. This action is called the conjugation actionof G on B. In particular, the group algebra kG itself is a left krG ˆ Gs-module and the blocks ofkG can be viewed as indecomposable krG ˆ Gs-modules under this action.


Notation 38.2Write ∆ : G ÝÑ G ˆ G, g ÞÑ pg, gq for the diagonal embedding of G in G ˆ G.

Theorem 38.3If B P BlppkGq, then every vertex of B, considered as an indecomposable krG ˆ Gs-module, hasthe form ∆pDq for some p-subgroup D ď G. Moreover, D is uniquely determined up to conjugationin G.

Proof : First, observe that G ˆ G permutes the k-basis of G consisting of the group elements and

StabGˆGp1q “ tpg1, g2q P G ˆ G | g1 ¨ 1 ¨ g´12 “ 1u “ ∆pGq .

It follows thatkG “ k ÒGˆG∆pGq

as a krGˆGs-module, and hence B is relatively ∆pGq-projective since it is by definition a direct summandof kG seen as krG ˆ Gs-module. Therefore, by Definition 26.2, a vertex Q of B (still considered as akrGˆGs-module) lies in ∆pGq. By Proposition 26.4, Q is a p-group, and thus there exists a p-subgroupD ď G such that Q “ ∆pDq, proving the first statement.Moreover, we know that ∆pDq is determined up to conjugacy in G ˆ G. Now, if D1 ď G is anotherp-subgroup such that ∆pD1q is a vertex of B, then there exists pg1, g2q P G ˆ G such that

∆pD1q “pg1,g2q∆pDq

and so for all x P D, pg1,g2qpx, xq “ p g1x, g2xq P ∆pD1q. Hence g1x P D1 for all x P D. Finally, as|D| “ |D1|, it follows that g1D “ D1.

Note: As a defect group of a block is uniquely determined up to G-conjugacy, it is clear that in factall defect groups have the same order.

Definition 38.4 (Defect group, defect)Let B P BlppkGq.

(a) A defect group of B is a p-subgroup D of G such that ∆pDq is a vertex of B considered as akrG ˆ Gs-module.

(b) If the defect groups of B have order pd (d P Zě0) then d is called the defect of B.

Defect groups are useful and important because in some sense they measure how far a p-block is frombeing semisimple.

Theorem 38.5If B is a block of kG with defect group D, then every indecomposable kG-module belonging to B isrelatively D-projective, and hence has a vertex contained in D.

Exercise 38.6Prove Theorem 38.5.[Hint: Prove that B, considered as kG-module via the conjugation action of G, is relatively D-projective.]


Corollary 38.7Let B be a block of kG with a trivial defect group. Then B is a simple algebra, and in particular, issemisimple.

Proof : If B has a trivial defect group D “ t1u, then by Theorem 38.5 every indecomposable kG-modulebelonging to B is t1u-projective, i.e. projective. So B is semisimple as all its modules are semisimple.But B is an indecomposable algebra by definition. Hence B is simple.

Main properties of the defect groups are the following. We mention them without proofs.

Theorem 38.8 (Green)Let B be a block of kG with defect group D. Then:

(a) if P is a Sylow p-subgroup of G containing D, then there exists g P CGpDq such thatD “ P X gP ;

(b) D contains every normal p-subgroup of G ;

(c) D is the largest normal p-subgroup of NGpDq, i.e. Q “ OppNGpQqq.

Example 19Let G be a p-group. We already saw that in this case BlppkGq “ tkGu. Then Theorem 38.8 showsthat this block has a unique defect group D “ G.

39 Brauer’s 1st and 2nd Main TheoremsFinally we present two main results due to Brauer.

Definition 39.1

Let H ď G, let b P BlppkHq. Then a block B P BlppkGq corresponds to b if and only if b | BÓGˆGHˆH ,and B is the unique block of kG with this property. We then write B “ bG . If such a block B exists,then we say that bG is defined.

Proposition 39.2 (Facts about bG )Let H ď G and let b be a block of kH with defect group D.

(a) If bG is defined, then D lies in a defect group of bG .

(b) If H ď N ď G, and bN , pbNqG and bG are defined, then bG “ pbNqG .

(c) If CGpDq ď H then bG is defined.


Theorem 39.3 (Brauer’s First Main Theorem)Let D ď G be a p-subgroup and let H ď G containing NGpDq. Then there is a bijection

tBlocks of kH with defect group Du „ÝÑ tBlocks of kG with defect group Du

b ÞÑ bG

Moreover, in this case bG is called the Brauer correspondent of b.

Proof (Sketch) : This is a particular case of the Green correspondence.

Theorem 39.4 (Brauer’s Second Main Theorem)Let H ď G, let B be a block of kG and let b be a block of kH . Suppose that V is an indecomposablemodule in B and U is an indecomposable module in b with vertex Q such that CGpQq ď H . If U isa direct summand of VÓGH , then bG is defined and bG “ B.

Lemma 39.5Let S be a simple kG-module. Then OppGq, the largest normal p-subgroup of G, acts trivially on S.In particular, the simple kG-modules are precisely the simple krG{OppGqs-modules inflated to kG.

Proof : Since OppGqE G, we know from Clifford’s Theorem that S ÓGOppGq is semisimple, hence of the form

S ÓGOppGq“ k ‘ ¨ ¨ ¨ ‘ k

since OppGq is a p-group and hence has only one simple module up to isomorphism, namely the trivialmodule. In other words, P acts trivially on S. The second claim follows immediately.

Corollary 39.6Let B be a block of kG with defect group D. Then there exists an indecomposable kG-modulebelonging to B with vertex D.

Proof : Write N :“ NGpDq. Let b P BlppkNq be a p-block with defect group D and let B P BlppkGq be theBrauer correspondent of b. As D is a defect group of b, D “ OppNq by Theorem 38.8(c). Now, let S bea simple kN-module belonging to b. Then, by Lemma 39.5, D acts trivially on S. So S can be viewedas a simple krN{Ds-module and we let PS be the projective cover of S seen as a krN{Ds-module. Then,by Corollary 36.4, the inflation InfNN{DpPSq of PS to N is an indecomposable kN-module belonging to b.

Claim 1: D is a vertex of InfNN{DpPSq.Indeed: By definition PS | krN{Ds, so

InfNN{DpPSq | InfNN{DpkrN{Dsq “ kÒND

and is therefore D-projective. Now, since DEN , it follows from Clifford’s Theorem that kÒNDÓND is a directsum of N-conjugates of the trivial kD-module k , which are all again trivial since D is a p-group. So

InfNN{DpPSqÓND | k ‘ ¨ ¨ ¨ ‘ k .

Therefore the indecomposable direct summands of InfNN{DpPSqÓND all have vertex D. It follows then fromLemma 27.1 that D is a vertex of InfNN{DpPSq, because InfNN{DpPSq ÓND has at least one indecomposabledirect summand with the same vertex as InfNN{DpPSq. This proves Claim 1.


Next we observe that the kG-Green correspondent fpInfNN{DpPSqq of InfNN{DpPSq is certainly an indecom-posable kG-module with vertex D.

Claim 2: fpInfNN{DpPSqq belongs to B.Indeed, this is clear by definition of the Green correspondent and Brauer’s First Main Theorem.

The next result shows that the converse of Corollary 38.7 also holds.

Corollary 39.7A block B of kG is a simple algebra if and only if B has a trivial defect group.

Proof : If B P BlppkGq has trivial defect group, then B is a simple algebra by Corollary 38.7. Suppose nowthat B is a block of kG which is a simple algebra. Then B is semisimple so all B-modules are projective.Hence all indecomposable B-modules have trivial vertices so, by Corollary 39.6, B has a trivial defectgroup.

Definition 39.8 (Principal block )

(a) The principal block of kG is the block of kG to which the trivial kG-module k belongs. Thisblock is denote by B0pkGq.

(b) A block of kG whose defect groups are the Sylow p-subgroups of G is said to have full defect .

Lemma 39.9

(a) Any indecomposable kG-module whose vertices are the Sylow p-subgroups of G belongs toa block of kG of full defect.

(b) The principal block B0pkGq is a block of full defect.

Proof : (a) is clear by Theorem 38.5.(b) is clear by (a) since the vertices of the trivial module are the Sylow p-subgroups of G.

Appendix 1: Background Material Module Theory

This appendix provides you with a short recap of the notions of the theory of modules, which we willassume as known for this lecture. We quickly review elementary definitions and constructions such asquotients, direct sum, direct products, tensor products and exact sequences, where we emphasise theapproach via universal properties.

Notation: throughout this appendix we let R and S denote rings, and unless otherwise specified, allrings are assumed to be unital and associative.

Most of the results are stated without proof, as they have been studied in the B.Sc. lecture CommutativeAlgebra. As further reference I recommend for example:

References:


A Modules, submodules, morphismsDefinition A.1 (Left R-module, right R-module, pR, Sq-bimodule)

(a) A left R-module is an ordered triple pM,`, ¨q, where pM,`q is an abelian group and¨ : R ˆM ÝÑ M,pr, mq ÞÑ r ¨m is a binary operation such that the map

λ : R ÝÑ EndpMqr ÞÑ λprq :“ λr : M ÝÑ M,m ÞÑ r ¨m

is a ring homomorphism. The operation ¨ is called a scalar multiplication or an externalcomposition law.

(b) A right R-module is defined analogously using a scalar multiplication ¨ : M ˆ R ÝÑ M,pm, rq ÞÑ m ¨ r on the right-hand side.

(c) An pR, Sq-bimodule is an abelian group pM,`q which is both a left R-module and a rightS-module, and which satisfies the axiom

r ¨ pm ¨ sq “ pr ¨mq ¨ s @ r P R,@ s P S,@m P M .

1000


Convention: Unless otherwise stated, in this lecture we always work with left modules. When noconfusion is to be made, we will simply write "R-module" to mean "left R-module", denote R-modulesby their underlying sets and write rm instead of r ¨m. Definitions for right modules and bimodules aresimilar to those for left modules, hence in the sequel we omit them.

Definition A.2 (R-submodule)An R-submodule of an R-module M is a subgroup U ď M such that r ¨ u P U @ r P R , @ u P U .

Definition A.3 (Morphisms)A (homo)morphism of R-modules (or an R-linear map, or an R-homomorphism) is a map of R-modules φ : M ÝÑ N such that:

(i) φ is a group homomorphism; and

(ii) φpr ¨mq “ r ¨ φpmq @ r P R , @ m P M .

Furthermore:

¨ An injective (resp. surjective) morphism of R-modules is sometimes called a monomorphism(resp. an epimorphism) and we often denote it with a hook arrow "ãÑ" (resp. a two-headarrow "�").

¨ A bijective morphism of R-modules is called an isomorphism (or an R-isomorphism), and wewrite M – N if there exists an R-isomorphism between M and N .

¨ A morphism from an R-module to itself is called an endomorphism and a bijective endomor-phism is called an automorphism .

Notation: We let RMod denote the category of left R-modules (with R-linear maps as morphisms),we let ModR denote the category of right R-modules (with R-linear maps as morphisms), and we letRModS denote the category of pR, Sq-bimodules (with pR, Sq-linear maps as morphisms).

Example A.4

(a) Exercise: Check that Definition A.1(a) is equivalent to requiring that pM,`, ¨q satisfies thefollowing axioms:

(M1) pM,`q is an abelian group;

(M2) pr1 ` r2q ¨m “ r1 ¨m` r2 ¨m for each r1, r2 P R and each m P M;

(M3) r ¨ pm1 `m2q “ r ¨m1 ` r ¨m2 for each r P R and all m1, m2 P M;

(M4) prsq ¨m “ r ¨ ps ¨mq for each r, s P R and all m P M .

(M5) 1R ¨m “ m for each m P M .

In other words, modules over rings satisfy the same axioms as vector spaces over fields. Hence:Vector spaces over a field K are K -modules, and conversely.


(b) Abelian groups are Z-modules, and conversely.Exercise: check it! What is the external composition law?

(c) If the ring R is commutative, then any right module can be made into a left module, andconversely.Exercise: check it! Where does the commutativity come into play?

(d) If φ : M ÝÑ N is a morphism of R-modules, then the kernel kerpφq :“ tm P M | φpmq “ 0Nuof φ is an R-submodule of M and the image Impφq :“ φpMq “ tφpmq | m P Mu of φ is anR-submodule of N .If M “ N and φ is invertible, then the inverse is the usual set-theoretic inverse map φ´1 andis also an R-homomorphism.Exercise: check it!

(e) Change of the base ring: if φ : S ÝÑ R is a ring homomorphism, then every R-module Mcan be endowed with the structure of an S-module with external composition law given by

¨ : S ˆM ÝÑ M, ps,mq ÞÑ s ¨m :“ φpsq ¨m .

Exercise: check it!

Notation A.5Given R-modules M and N , we set HomRpM,Nq :“ tφ : M ÝÑ N | φ is an R-homomorphismu.This is an abelian group for the pointwise addition of maps:

` : HomRpM,Nq ˆHomRpM,Nq ÝÑ HomRpM,Nqpφ, ψq ÞÑ φ ` ψ : M ÝÑ N,m ÞÑ φpmq ` ψpmq .

In case N “ M , we write EndRpMq :“ HomRpM,Mq for the set of endomorphisms of M andAutRpMq for the set of automorphisms of M , i.e. the set of invertible endomorphisms of M .

Lemma-Definition A.6 (Quotients of modules)Let U be an R-submodule of an R-module M . The quotient group M{U can be endowed with thestructure of an R-module in a natural way via the external composition law

R ˆM{U ÝÑ M{U`

r, m` U˘

ÞÝÑ r ¨m` U

The canonical map π : M ÝÑ M{U,m ÞÑ m ` U is R-linear and we call it the canonical (ornatural) homomorphism.

Definition A.7 (Cokernel, coimage)Let φ P HomRpM,Nq. The cokernel of φ is the quotient R-module cokerpφq :“ N{ Imφ, and thecoimage of φ is the quotient R-module M{ kerφ.

Theorem A.8 (The universal property of the quotient and the isomorphism theorems)

(a) Universal property of the quotient: Let φ : M ÝÑ N be a homomorphism of R-modules.


If U is an R-submodule of M such that U Ď kerpφq, then there exists a unique R-modulehomomorphism φ : M{U ÝÑ N such that φ ˝π “ φ, or in other words such that the followingdiagram commutes:

M N

M{U

π

φ

ö

D!φ

Concretely, φpm` Uq “ φpmq @ m` U P M{U .

(b) 1st isomorphism theorem: With the notation of (a), if U “ kerpφq, then

φ : M{ kerpφq ÝÑ Impφq

is an isomorphism of R-modules.

(c) 2nd isomorphism theorem: If U1, U2 are R-submodules of M , then so are U1XU2 and U1Ù2,and there is an isomorphism of R-modules

pU1 ` U2q{U2 – U1{pU1 X U2q .

(d) 3rd isomorphism theorem: If U1 Ď U2 are R-submodules of M , then there is an isomorphismof R-modules

pM{U1q {pU2{U1q – M{U2 .

(e) Correspondence theorem: If U is an R-submodule of M , then there is a bijection

tR-submodules X of M | U Ď Xu ÐÑ tR-submodules of M{UuX ÞÑ X{U

π´1pZ q Ð[ Z .

B Free modules and projective modules

Free modules

Definition B.1 (Generating set / R-basis / finitely generated/free R-module)Let M be an R-module and let X Ď M be a subset. Then:

(a) M is said to be generated by X if every element m P M may be written as an R-linearcombination m “

ř

xPX λxx , i.e. where λx P R is almost everywhere 0. In this case we writeM “ xXyR or M “

ř

xPX Rx .

(b) M is said to be finitely generated if it admits a finite set of generators.

(c) X is an R-basis (or simply a basis) if X generates M and if every element of M can be writtenin a unique way as an R-linear combination

ř

xPX λxX (i.e. with λx P R almost everywhere 0).


(d) M is called free if it admits an R-basis X , and |X | is called the R-rank of M .Notation: In this case we write M “

À

xPX Rx –À

xPX R .

Remark B.2

(a) Warning: If the ring R is not commutative, then it is not true in general that two differentbases of a free R-module have the same number of elements.

(b) Let X be a generating set for M . Then, X is a basis of M if and only if S is R-linearlyindependent.

(c) If R is a field, then every R-module is free. (R-modules are R-vector spaces in this case!)

Proposition B.3 (Universal property of free modules)Let M be a free R-module with R-basis X . If N is an R-module and f : X ÝÑ N is a map (ofsets), then there exists a unique R-homomorphism pf : M ÝÑ N such that the following diagramcommutes:

X N

Minc

f

ö

D!pf

We say that pf is obtained by extending f by R-linearity.

Proof : Given an R-linear combinationř

xPX λxx P M , set pfpř

xPX λxxq :“ř

xPX λxfpxq. The claim follows.

Proposition B.4 (Properties of free modules)

(a) Every R-module M is isomorphic to a quotient of a free R-module.

(b) If P is a free R-module, then HomRpP,´q is an exact functor.

Projective modules

Proposition-Definition B.5 (Projective module)Let P be an R-module. Then the following are equivalent:

(a) The functor HomRpP,´q is exact.

(b) If ψ P HomRpM,Nq is a surjective morphism of R-modules, then the morphism of abeliangroups ψ˚ : HomRpP,Mq ÝÑ HomRpP,Nq is surjective.

(c) If π : M ÝÑ P is a surjective R-linear map, then π splits, i.e., there exists σ : P ÝÑ M suchthat π ˝ σ “ IdP .

(d) P is isomorphic to a direct summand of a free R-module.


If P satisfies these equivalent conditions, then P is called projective.

Example B.6

(a) If R “ Z, then every submodule of a free Z-module is again free (main theorem on Z-modules).

(b) Let e be an idempotent in R , that is e2 “ e. Then, R – Re‘ Rp1´ eq and Re is projectivebut not free if e ‰ 0, 1.

(d) A direct sum of modulesÀ

iPI Pi is projective if and only if each Pi is projective.

C Direct products and direct sumsLet tMiuiPI be a family of R-modules. Then the abelian group

ś

iPIMi, that is the product of tMiuiPIseen as a family of abelian groups, becomes an R-module via the following external composition law:

R ˆź

iPIMi ÝÑ

ź

iPIMi

`

r, pmiqiPI˘

ÞÝÑ`

r ¨mi˘

iPI .

Furthermore, for each j P I , we let πj :ś

iPIMi ÝÑ Mj , pmiqiPI ÞÑ mj denotes the j-th projection fromthe product to the module Mj .

Proposition C.1 (Universal property of the direct product)If tφi : L ÝÑ MiuiPI is a family of R-homomorphisms, then there exists a unique R-homomorphismφ : L ÝÑ

ś

iPIMi such that πj ˝ φ “ φj for every j P I .

L

¨¨¨ φkö

φj ¨¨¨ö

��

φ

��ś

iPIMi

πk{{πj

##Mk Mj

Thus,

HomR

´

L,ź

iPIMi

¯

ÝÑź

iPIHomRpL,Miq

f ÞÝÑ`

πi ˝ f˘

iPI

is an isomorphism of abelian groups.

Now letÀ

iPIMi be the subgroup ofś

iPIMi consisting of the elements pmiqiPI such that mi “ 0 al-most everywhere (i.e. mi “ 0 exept for a finite subset of indices i P I). This subgroup is called thedirect sum of the family tMiuiPI and is in fact an R-submodule of the product. For each j P I , we letηj : Mj ÝÑ

À

iPIMi, mj ÞÑ denote the canonical injection of Mj in the direct sum.


Proposition C.2 (Universal property of the direct sum)If tfi : Mi ÝÑ LuiPI is a family of R-homomorphisms, then there exists a unique R-homomorphismφ :

À

iPIMi ÝÑ L such that f ˝ ηj “ fj for every j P I .

L

À

iPIMi

f

OO

Mk

ηk

;;

¨¨¨ fkö

CC

Mj

ηj

cc

fj ¨¨¨ö

[[

Thus,

HomR

´

à

iPIMi, L

¯

ÝÑź

iPIHomRpMi, Lq

f ÞÝÑ`

f ˝ ηi˘

iPI

is an isomorphism of abelian groups.

Remark C.3It is clear that if |I| ă 8, then

À

iPIMi “ś

iPIMi.

The direct sum as defined above is often called an external direct sum. This relates as follows with theusual notion of internal direct sum:

Definition C.4 (“Internal” direct sums)Let M be an R-module and N1, N2 be two R-submodules of M . We write M “ N1 ‘N2 if everym P M can be written in a unique way as m “ n1 ` n2, where n1 P N1 and n2 P N2.

In fact M “ N1 ‘N2 (internal direct sum) if and only if M “ N1 `N2 and N1 XN2 “ t0u.

Proposition C.5If N1, N2 and M are as above and M “ N1 ‘N2 then the homomorphism of R-modules

φ : M ÝÑ N1 ˆN2 “ N1 ‘N2 (external direct sum)m “ n1 ` n2 ÞÑ pn1, n2q ,

is an isomorphism of R-modules.

The above generalises to arbitrary internal direct sums M “À

iPI Ni.

D Exact sequencesExact sequences constitute a very useful tool for the study of modules. Often we obtain valuable infor-mation about modules by plugging them in short exact sequences, where the other terms are known.


Definition D.1 (Exact sequence)

A sequence L φÝÑ M ψ

ÝÑ N of R-modules and R-linear maps is called exact (at M) if Imφ “ kerψ.

Remark D.2 (Injectivity/surjectivity/short exact sequences)

(a) L φÝÑ M is injective ðñ 0 ÝÑ L φ

ÝÑ M is exact at L.

(b) M ψÝÑ N is surjective ðñ M ψ

ÝÑ N ÝÑ 0 is exact at N .

(c) 0 ÝÑ L φÝÑ M ψ

ÝÑ N ÝÑ 0 is exact (i.e. at L, M and N) if and only if φ is injective, ψ issurjective and ψ induces an R-isomorphism ψ : M{ Imφ ÝÑ N,m` Imφ ÞÑ ψpmq.Such a sequence is called a short exact sequence (s.e.s. for short).

(d) If φ P HomRpL,Mq is an injective morphism, then there is a s.e.s.

0 ÝÑ L φÝÑ M π

ÝÑ cokerpφq ÝÑ 0

where π is the canonical projection.

(e) If ψ P HomRpM,Nq is a surjective morphism, then there is a s.e.s.

0 ÝÑ kerpψq iÝÑ M ψ

ÝÑ N ÝÑ 0 ,

where i is the canonical injection.

Proposition D.3Let Q be an R-module. Then the following holds:

(a) HomRpQ,´q : RMod ÝÑ Ab is a left exact covariant functor. In other words, if0 ÝÑ L φ

ÝÑ M ψÝÑ N ÝÑ 0 is a s.e.s of R-modules, then the induced sequence

0 // HomRpQ, Lqφ˚ // HomRpQ,Mq

ψ˚ // HomRpQ,Nq

is an exact sequence of abelian groups. Here φ˚ :“ HomRpQ,φq, that is φ˚pαq “ φ ˝ α forevery α P HomRpQ, Lq and similarly for ψ˚.

(b) HomRp´, Qq : RMod ÝÑ Ab is a left exact contravariant functor. In other words, if0 ÝÑ L φ

ÝÑ M ψÝÑ N ÝÑ 0 is a s.e.s of R-modules, then the induced sequence

0 // HomRpN,Qqψ˚ // HomRpM,Qq

φ˚ // HomRpL,Qq

is an exact sequence of abelian groups. Here φ˚ :“ HomRpφ,Qq, that is φ˚pαq “ α ˝ φ forevery α P HomRpM,Qq and similarly for ψ˚.

Exercise: verify that HomRpQ,´q and HomRp´, Qq are functors.Notice that HomRpQ,´q and HomRp´, Qq are not right exact in general. Exercise: find counter-examples!


Lemma-Definition D.4 (Split short exact sequence)

A s.e.s. 0 ÝÑ L φÝÑ M ψ

ÝÑ N ÝÑ 0 of R-modules is called split if it satisfies one of the followingequivalent conditions:

(a) ψ admits an R-linear section, i.e. if D σ P HomRpN,Mq such that ψ ˝ σ “ IdN ;

(b) φ admits an R-linear retraction, i.e. if D ρ P HomRpM, Lq such that ρ ˝ φ “ IdL;

(c) D an R-isomorphism α : M ÝÑ L‘N such that the following diagram commutes:

0 // Lφ //

IdL��

ö

Mψ //

α��

ö

N //

IdN��

0

0 // L i // L‘Np // N // 0 ,

where i, resp. p, are the canonical inclusion, resp. projection.

Remark D.5If the sequence splits and σ is a section, then M “ φpLq ‘ σpNq. If the sequence splits and ρ is aretraction, then M “ φpLq ‘ kerpρq.

Example D.6The s.e.s. of Z-modules

0 // Z{2Zφ // Z{2Z‘ Z{2Z π // Z{2Z // 0

defined by φpr1sq “ pr1s, r0sq and where π is the canonical projection onto the cokernel of φ issplit but the sequence

0 // Z{2Zφ // Z{4Z π // Z{2Z // 0

defined by φpr1sq “ pr2sq and π is the canonical projection onto the cokernel of φ is not split.Exercise: justify this fact using a straightforward argument.

E Tensor productsDefinition E.1 (Tensor product of R-modules)

Let M be a right R-module and let N be a left R-module. Let F be the free abelian group (= freeZ-module) with basis M ˆN . Let G be the subgroup of F generated by all the elements

pm1 `m2, nq ´ pm1, nq ´ pm2, nq, @m1, m2 P M,@n P N,pm,n1 ` n2q ´ pm,n1q ´ pm,n2q, @m P M,@n1, n2 P N, andpmr, nq ´ pm, rnq, @m P M,@n P N,@r P R.


The tensor product of M and N (balanced over R ), is the abelian group M bR N :“ F{G . Theclass of pm,nq P F in M bR N is denoted by mb n.

Remark E.2

(a) M bR N “ xmb n | m P M,n P NyZ.

(b) In M bR N , we have the relations

pm1 `m2q b n “ m1 b n`m2 b n, @m1, m2 P M,@n P N,mb pn1 ` n2q “ mb n1 `mb n2, @m P M,@n1, n2 P N, andmr b n “ mb rn, @m P M,@n P N,@r P R.

In particular, mb 0 “ 0 “ 0b n @ m P M , @ n P N and p´mq b n “ ´pmb nq “ mb pńq@ m P M , @ n P N .

Definition E.3 (R-balanced map)Let M and N be as above and let A be an abelian group. A map f : M ˆ N ÝÑ A is calledR-balanced if

fpm1 `m2, nq “ fpm1, nq ` fpm2, nq, @m1, m2 P M,@n P N,fpm,n1 ` n2q “ fpm,n1q ` fpm,n2q, @m P M,@n1, n2 P N,fpmr, nq “ fpm, rnq, @m P M,@n P N,@r P R.

Remark E.4The canonical map t : M ˆN ÝÑ M bR N, pm,nq ÞÑ mb n is R-balanced.

Proposition E.5 (Universal property of the tensor product)Let M be a right R-module and let N be a left R-module. For every abelian group A and everyR-balanced map f : M ˆN ÝÑ A there exists a unique Z-linear map f : M bR N ÝÑ A such thatthe following diagram commutes: M ˆN f //

t��

A

M bR Nf

ö

;;

Proof : Let ı : M ˆ N ÝÑ F denote the canonical inclusion, and let π : F ÝÑ F{G denote the canonicalprojection. By the universal property of the free Z-module, there exists a unique Z-linear map f : F ÝÑ Asuch that f ˝ ı “ f . Since f is R-balanced, we have that G Ď kerpfq. Therefore, the universal property ofthe quotient yields the existence of a unique homomorphism of abelian groups f : F{G ÝÑ A such thatf ˝ π “ f :

M ˆN f //

ı��

t

A

F

f

::

π��

M bR N – F{Gf

JJ

Clearly t “ π ˝ ı, and hence f ˝ t “ f ˝ π ˝ ı “ f ˝ ı “ f .


Remark E.6Let M and N be as in Definition E.1.

(a) Let tMiuiPI be a collection of right R-modules, M be a right R-module, N be a left R-moduleand tNjuiPJ be a collection of left R-modules. Then, we have

à

iPIMi bR N –

à

iPIpMi bR Nq

M bRà

jPJNj –

à

jPJpM bR Njq.

(This is easily proved using both the universal property of the direct sum and of the tensorproduct.)

(b) There are natural isomorphisms of abelian groups given by R bR N – N via rbn ÞÑ rn, andM bR R – M via mb r ÞÑ mr.

(c) It follows from (b), that if P is a free left R-module with R-basis X , then N bR P –À

xPX N ,and if P is a free right R-module with R-basis X , then P bR M –

À

xPX M .

(d) Let Q be a third ring. Then we obtain module structures on the tensor product as follows:

(i) If M is a pQ,Rq-bimodule and N a left R-module, then M bR N can be endowed withthe structure of a left Q-module via

q ¨ pmb nq “ q ¨mb n @q P Q,@m P M,@n P N.

(ii) If M is a right R-module and N an pR, Sq-bimodule, then MbR N can be endowed withthe structure of a right S-module via

pmb nq ¨ s “ qmb n ¨ s @s P S,@m P M,@n P N.

(iii) If M is a pQ,Rq-bimodule and N an pR, Sq-bimodule. Then M bR N can be endowedwith the structure of a pQ,Sq-bimodule via the external composition laws defined in (i)and (ii).

(e) Assume R is commutative. Then any R-module can be viewed as an pR,Rq-bimodule. Then,in particular, M bR N becomes an R-module (both on the left and on the right).

(f ) For instance, it follows from (e) that if K is a field and M and N are K -vector spaces withK -bases txiuiPI and tyjujPJ resp., then M bK N is a K -vector space with a K -basis given bytxi b yjupi,jqPIˆJ .

(g) Tensor product of morphisms: Let f : M ÝÑ M 1 be a morphism of right R-modules andg : N ÝÑ N 1 be a morphism of left R-modules. Then, by the universal property of thetensor product, there exists a unique Z-linear map f b g : M bR N ÝÑ M 1 bR N 1 such thatpf b gqpmb nq “ fpmq b gpnq.


Exercise E.7

(a) Assume R is a commutative ring and I is an ideal of R . Let M be a left R-module. Prove thatthere is an isomorphism of left R-modules R{I bR M – M{IM .

(b) Let m,n be coprime positive integers. Compute Z{nZbZ Z{mZ, QbZ Q, and Q{ZbZ Q.

(c) Let K be a field and let U,V be finite-dimensional K -vector spaces. Prove that there is anatural isomorphism of K -vector spaces:

HomK pU,V q – U˚ bK V .

Proposition E.8 (Right exactness of the tensor product)

(a) Let N be a left R-module. Then ´bR N : ModR ÝÑ Ab is a right exact covariant functor.

(b) Let M be a right R-module. Then M bR ´ :R Mod ÝÑ Ab is a right exact covariant functor.

Remark E.9The functors ´bR N and M bR ´ are not left exact in general.

F AlgebrasIn this lecture we aim at studying modules over specific rings, which are in particular algebras.

Definition F.1 (Algebra)Let R be a commutative ring.

(a) An R-algebra is an ordered quadruple pA,`, ¨, ˚q such that the following axioms hold:

(A1) pA,`, ¨q is a ring;(A2) pA,`, ˚q is a left R-module; and(A3) r ˚ pa ¨ bq “ pr ˚ aq ¨ b “ a ¨ pr ˚ bq @ a, b P A, @ r P R .

(b) A map f : AÑ B between two R-algebras is called an algebra homomorphism iff:

(i) f is a homomorphism of R-modules;(ii) f is a ring homomorphism.

Example F.2 (Algebras)

(a) The ring R itself is an R-algebra.[The internal composition law "¨" and the external composition law "˚" coincide in this case.]

(b) For each n P Zě1 the set MnpRq of nˆ n-matrices with coefficients in R is an R-algebra forits usual R-module and ring structures.


[Note: in particular R-algebras need not be commutative rings in general!]

(c) Let K be a field. Then for each n P Zě1 the polynom ring K rX1, . . . , Xns is a K -algebra forits usual K -vector space and ring structure.

(d) R and C are Q-algebras, C is an R-algebra, . . .

(e) Rings are Z-algebras.Exercise: Check it!

Example F.3 (Modules over algebras)

(a) A “ MnpRq ñ Rn is an A-module for the external composition law given by left matrixmultiplication Aˆ Rn ÝÑ Rn, pB, xq ÞÑ Bx .

(b) If K is a field and V a K -vector space, then V becomes an A-algebra for A :“ EndK pV qtogether with the external composition law

Aˆ V ÝÑ V , pφ, vq ÞÑ φpvq .

Exercise: Check it!

(c) An arbitrary A-module M can be seen as an R-module via a change of the base ring sinceR ÝÑ A, r ÞÑ r ˚ 1A is a homomorphism of rings by the algebra axioms.

Exercise F.4Let R be a commutative ring.

(a) Let M,N be R-modules. Prove that:

(1) EndRpMq, endowed with the pointwise addition of maps and the usual composition ofmaps, is a ring. (Note that the commutativity of R is not necessary!)

(2) The abelian group HomRpM,Nq is a left R-module for the external composition lawdefined by

prfqpmq :“ fprmq “ rfpmq @ r P R, @f P HomRpM,Nq, @m P M .

(b) Let now A be an R-algebra and M be an A-module. Prove that EndRpMq and EndApMq areR-algebras.

Appendix 2: The Language of Category Theory

This appendix gives a short introduction to some of the basic notions of category theory used in thislecture.

References:

[Mac98] S. Mac Lane. Categories for the working mathematician. Second. Vol. 5. Springer-Verlag,New York, 1998.

[Wei94] C. A. Weibel. An introduction to homological algebra. Vol. 38. Cambridge Studies in AdvancedMathematics. Cambridge University Press, Cambridge, 1994.

G CategoriesDefinition G.1 (Category)

A category C consists of:‚ a class Ob C of objects,

‚ a set HomCpA,Bq of morphisms for every ordered pair pA,Bq of objects, and

‚ a composition function

HomCpA,Bq ˆHomCpB,Cq ÝÑ HomCpA, Cqpf , gq ÞÑ g ˝ f

for each ordered triple pA,B, Cq of objects,satisfying the following axioms:(C1) Unit axiom: for each object A P Ob C, there exists an identity morphism 1A P HomCpA, Aq

such that for every f P HomCpA,Bq for all B P Ob C,f ˝ 1A “ f “ 1B ˝ f .

(C2) Associativity axiom: for every f P HomCpA,Bq, g P HomCpB,Cq and h P HomCpC,Dq withA,B, C,D P Ob C,

h ˝ pg ˝ fq “ ph ˝ gq ˝ f .

1013


Let us start with some remarks and examples to enlighten this definition:Remark G.2

(a) Ob C need not be a set!

(b) The only requirement on HomCpA,Bq is that it be a set, and it is allowed to be empty.

(c) It is common to write f : A ÝÑ B or A fÝÑ B instead of f P HomCpA,Bq, and to talk about

arrows instead of morphisms. It is also common to write "A P C" instead of "A P Ob C".

(d) The identity morphism 1A P HomCpA, Aq is uniquely determined: indeed, if fA P HomCpA, Aqwere a second identity morphisms, then we would have fA “ fA ˝ 1A “ 1A.

Example G.3

(a) C “ 1 : category with one object and one morphism (the identity morphism):

‚

1‚

(b) C “ 2 : category with two objects and three morphism, where two of them are identitymorphisms and the third one goes from one object to the other:

A B1A 1B

(c) A group G can be seen as a category CpGq with one object: Ob CpGq “ t‚u, HomCpGqp‚, ‚q “ G(notice that this is a set) and composition is given by multiplication in the group.

(d) The nˆm-matrices with entries in a field k for n,m ranging over the positive integers forma category Matk : Ob Matk “ Zą0, morphisms n ÝÑ m from n to m are the mˆ n-matrices,and compositions are given by the ordinary matrix multiplication.

Example G.4 (Categories and algebraic structures)

(a) C “ Set, the category of sets: objects are sets, morphisms are maps of sets, and compositionis the usual composition of functions.

(b) C “ Veck , the category of vector spaces over the field k : objects are k-vector spaces, mor-phisms are k-linear maps, and composition is the usual composition of functions.

(c) C “ Top, the category of topological spaces: objects are topological spaces, morphisms arecontinous maps, and composition is the usual composition of functions.

(d) C “ Grp, the category of groups: objects are groups, morphisms are homomorphisms of groups,and composition is the usual composition of functions.


(e) C “ Ab, the category of abelian groups: objects are abelian groups, morphisms are homomor-phisms of groups, and composition is the usual composition of functions.

(f ) C “ Rng, the category of rings: objects are rings, morphisms are homomorphisms of rings,and composition is the usual composition of functions.

(g) C “R Mod, the category of left R-modules: objects are left modules over the ring R , morphismsare R-homomorphisms, and composition is the usual composition of functions.

(g’) C “ ModR , the category of left R-modules: objects are right modules over the ring R ,morphisms are R-homomorphisms, and composition is the usual composition of functions.

(g”) C “R ModS , the category of pR, Sq-bimodules: objects are pR, Sq-bimodules over the ringsR and S, morphisms are pR, Sq-homomorphisms, and composition is the usual composition offunctions.

(h) Examples of your own . . .

Definition G.5 (Monomorphism/epimorphism)Let C be a category and let f P HomCpA,Bq be a morphism. Then f is called

(a) a monomorphism iff for all morphisms g1, g2 : C ÝÑ A,

f ˝ g1 “ f ˝ g2 ùñ g1 “ g2 .

(b) an epimorphism iff for all morphisms g1, g2 : B ÝÑ C ,

g1 ˝ f “ g2 ˝ f ùñ g1 “ g2 .

Remark G.6In categories, where morphisms are set-theoretic maps, then injective morphisms are monomorphisms,and surjective morphisms are epimorphisms.In module categories (RMod, ModR , RModS , . . . ), the converse holds as well, but:Warning: It is not true in general, that all monomorphisms must be injective, and all epimorphismsmust be surjective.For example in Rng, the canonical injection ı : Z ÝÑ Q is an epimorphism. Indeed, if C is a ringand g1, g2 P HomRngpQ, Cq

Z ı // Qg1//

g2 // C

are such that g1 ˝ ı “ g2 ˝ ı, then we must have g1 “ g2 by the universal property of the field offractions. However, ı is clearly not surjective.


H FunctorsDefinition H.1 (Covariant functor )

Let C and D be categories. A covariant functor F : C ÝÑ D is a collection of maps:

‚ F : Ob C ÝÑ Ob D , X ÞÑ F pXq, and

‚ FA,B : HomCpA,Bq ÞÑ HomD pF pAq, F pBqq,

satisfying:

(a) If A fÝÑ B g

ÝÑ C are morphisms in C, then F pg ˝ fq “ F pgq ˝ F pfq; and

(b) F p1Aq “ 1FpAq for every A P Ob C.

Definition H.2 (Contravariant functor )Let C and D be categories. A contravariant functor F : C ÝÑ D is a collection of maps:

‚ F : Ob C ÝÑ Ob D , X ÞÑ F pXq, and

‚ FA,B : HomCpA,Bq ÞÑ HomD pF pBq, F pAqq,

satisfying:

(a) If A fÝÑ B g

ÝÑ C are morphisms in C, then F pg ˝ fq “ F pfq ˝ F pgq; and

(b) F p1Aq “ 1FpAq for every A P Ob C.

Remark H.3Often in the literature functors are defined only on objects of categories. When no confusion is tobe made and the action of functors on the morphism sets are implicitely obvious, we will also adoptthis convention.

Example H.4Let Q P ObpRModq. Then

HomRpQ,´q : RMod ÝÑ AbM ÞÑ HomRpQ,Mq ,

is a covariant functor, and

HomRp´, Qq : RMod ÝÑ AbM ÞÑ HomRpM,Qq ,

is a contravariant functor.

Exact Functors.We are now interested in the relations between functors and exact sequences in categories where itmakes sense to define exact sequences, that is categories that behave essentially like module categories


such as RMod. These are the so-called abelian categories. It is not the aim, to go into these details,but roughly speaking abelian categories are categories satisfying the following properties:

‚ they have a zero object (in RMod: the zero module)

‚ they have products and coproducts (in RMod: products and direct sums)

‚ they have kernels and cokernels (in RMod: the usual kernels and cokernels of R-linear maps)

‚ monomorphisms are kernels and epimorphisms are cokernels (in RMod: satisfied)

Definition H.5 (Pre-additive categories/additive functors)

(a) A category C in which all sets of morphisms are abelian groups is called pre-additive.

(b) A functor F : C ÝÑ D between pre-additive categories is called additive iff the maps FA,Bare homomorphisms of groups for all A,B P Ob C.

Definition H.6 (Left exact/right exact/exact functors)Let F : C ÝÑ D be a covariant (resp. contravariant) additive functor between two abelian categories,and let 0 ÝÑ A f

ÝÑ B gÝÑ C ÝÑ 0 be a s.e.s. of objects and morphisms in C. Then F is called:

(a) left exact if 0 ÝÑ F pAq FpfqÝÑ F pBq FpgqÝÑ F pCq (resp. 0 ÝÑ F pCq FpgqÝÑ F pBq FpfqÝÑ F pAqq) is anexact sequence.

(b) right exact if F pAq FpfqÝÑ F pBq FpgqÝÑ F pCq ÝÑ 0 (resp. F pCq FpgqÝÑ F pBq FpfqÝÑ F pAqq ÝÑ 0) is anexact sequence.

(c) exact if 0 ÝÑ F pAq FpfqÝÑ F pBq FpgqÝÑ F pCq ÝÑ 0 (resp. 0 ÝÑ F pCq FpgqÝÑ F pBq FpfqÝÑF pAqq ÝÑ0)is a short exact sequence.

Example H.7The functors HomRpQ,´q and HomRp´, Qq of Example H.4 are both left exact functors. MoreoverHomRpQ,´q is exact if and only if Q is projective, and HomRp´, Qq is exact if and only if Q isinjective.

Index of Notation

General symbolsC field of complex numbersFq finite field with q elementsIdM identity map on the set MImpfq image of the map fkerpφq kernel of the morphism φN the natural numbers without 0N0 the natural numbers with 0O discrete valuation ringP the prime numbers in ZQ field of rational numbersQp field of p-adic numbersR field of real numbersZ ring of integer numbersZěa,Ząa,Zďa,Zăa tm P Z | m ě a (resp. m ą a,m ě a,m ă aquZp ring of p-adic integers|X | cardinality of the set Xδij Kronecker’s deltaŤ

unionš

disjoint unionŞ

intersectionř

summation symbolś

, ˆ cartesian/direct product¸ semi-direct product‘ direct sumb tensor productH empty set@ for allD there exists– isomorphisma | b , a - b a divides b, a does not divide bpa, bq gcd of a and bpF,O, kq p-modular systemf |S restriction of the map f to the subset SãÑ injective map� surjective map

Group theoryAutpGq automorphism group of the group GAn alternating group on n lettersCm cyclic group of order m in multiplicative notationCGpxq centraliser of the element x in GCGpHq centraliser of the subgroup H in GD2n dihedral group of order 2nδ : G Ñ G ˆ G diagonal mapEndpAq endomorphism ring of the abelian group AG{N quotient group G modulo NGLnpK q general linear group over KHgL pH, Lq-double cosetrHzG{Ls set of pH, Lq-double coset representativesH ď G, H ă G H is a subgroup of G, resp. a proper subgroupN Ĳ G N is a normal subgroup GNGpHq normaliser of H in GN ¸θ H semi-direct product of N in H w.r.t. θSn symmetric group on n lettersSLnpK q special linear group over KZ{mZ cyclic group of order m in additive notationxg conjugate of g by x , i.e. gxg´1

xgy Ď G subgroup of G generated by g|G : H| index of the subgroup H in GrG{Hs set of left coset representatives of Hx P G{N class of x P G in the quotient group G{Nt1u, 1 trivial group

Module theoryHomRpM,Nq R-homomorphisms from M to NEndRpMq R-endomorphism ring of the R-module MhdpMq head of the module MKG group algebra of the group G over the commutative ring Kε : KG ÝÑ K augmentation mapIpKGq augmentation idealJpRq Jacobson radical of the ring RM | N M is a direct summand of NM bR N tensor product of M and N balanced over RMG G-fixed points of the module MMG G-cofixed points of the module MM ÓGH , ResGHpMq restriction of M from G to HM ÒHG , IndGHpMq induction of M from H to GInfGG{NpMq inflation of M from G{N to GRˆ units of the ring RR˝ regular left R-module on the ring RradpMq radical of the module MsocpMq socle of the module M

xXyR R-module generated by the set XV F extension of scalars F bO VZ(R) centre of the ring R

Character and Block TheorybG Brauer correspondent of bC Cartan matrix of GClF pGq, ClF pGp1q the class functions on G or Gp1DecppGq decomposition matrixGp1 p-regular elements of GIrrF pGq ordinary irreducible F-characters of GIBrppGq irreducible p-Brauer characters of Gχreg regular characterρreg regular representationΦφ projective indecomposable character associated to φ P IBrpGq

Category TheoryOb C objects of the category CHomCpA,Bq morphisms from A to BSet the category of setsVeck the category of vector spaces over the field kTop the category of topological spacesGrp the category of groupsAb the category of abelian groupsRng the category of ringsRMod the category of left R-modulesModR the category of left R-modulesRModS the category of pR, Sq-bimodules

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