Brauer Algebras of Non-simply Laced TypeBrauer Algebras of Non-simply Laced Type PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag

Brauer algebras of non-simply laced type

Citation for published version (APA):Liu, S. (2012). Brauer algebras of non-simply laced type. Eindhoven: Technische Universiteit Eindhoven.https://doi.org/10.6100/IR732986

DOI:10.6100/IR732986

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https://research.tue.nl/en/publications/brauer-algebras-of-nonsimply-laced-type(65b08b51-65d0-4685-87f4-fdb292f0d246).html

Brauer Algebras of Non-simply Laced Type

Shoumin Liu

A catalogue record is available from the Eindhoven University of Technology Library.ISBN: 978-90-386-3161-5

Brauer Algebras of Non-simply Laced Type

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan deTechnische Universiteit Eindhoven, op gezag van derector magnificus, prof.dr.ir. C.J. van Duijn, voor een

commissie aangewezen door het College voorPromoties in het openbaar te verdedigenop woensdag 6 juni 2012 om 16.00 uur

door

Shoumin Liu

geboren te Shandong, China

Dit proefschrift is goedgekeurd door de promotor:

prof.dr. A. M. Cohen

CONTENTS

0 Introduction 10.1 A little history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 The importance of Brauer algebras . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Diagram algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4 The aims and results of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . 2

1 Preliminary knowledge 51.1 Coxeter groups and Weyl groups . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 A poset of simply laced type . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Brauer algebras of simply laced types . . . . . . . . . . . . . . . . . . . . . . . 91.4 Cellular algebra and cellularly stratified algebra . . . . . . . . . . . . . . . . 101.5 The classical Brauer algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.6 Definition of Brauer algebras of type Bn, Cn, F4, G2 . . . . . . . . . . . . . . 171.7 BMW algebras of simply laced type . . . . . . . . . . . . . . . . . . . . . . . . 23

2 Type Cn 252.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2 The homomorphism φ and the number an . . . . . . . . . . . . . . . . . . . . 272.3 Elementary properties of type C algebras . . . . . . . . . . . . . . . . . . . . . 292.4 Surjectivity of φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.5 Admissible sets and their orbits . . . . . . . . . . . . . . . . . . . . . . . . . . 362.6 Further properties of type C algebras . . . . . . . . . . . . . . . . . . . . . . . 46

3 Type Bn 493.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2 Elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.3 Admissible root sets and the monoid action . . . . . . . . . . . . . . . . . . . 593.4 An upper bound on the rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.5 Cellularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4 Type F4 774.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.2 Basic properties of Br(F4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.3 The root system of type F4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.4 An upper bound for the rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.5 φ(Br(F4)) in Br(E6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

vi CONTENTS

4.6 Cellularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5 Type In2 87

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.2 An interesting elementary problem . . . . . . . . . . . . . . . . . . . . . . . . 895.3 Normal forms for BrM(In

2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.4 The rank of Imφ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6 Types H3 and type H4 976.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986.3 Admissible root sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986.4 Normal forms for BrM(H3) and BrM(H4) . . . . . . . . . . . . . . . . . . . . . 1016.5 Images of φ1 and φ2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7 Type G2 1057.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057.2 Root systems for D4 and G2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067.3 The map φ inducing a homomorphism . . . . . . . . . . . . . . . . . . . . . . 1067.4 Normal forms of BrM(G2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077.5 The algebra SBr(D4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Bibliography 111

Index 114

Acknowledgements 115

Curriculum Vitae 117

Summary:Brauer algebras ofnon-simply laced type 119

0INTRODUCTION

0.1 A little history

In the history and family of mathematics, group theory is a quite young child, comparedwith geometry and number theory, or other brothers. It was first employed by the youngtalented mathematician Galois to solve the problems of finding zeros of polynomials. Af-ter this, there came many pioneers in this field, such as Cauchy and Abel.For groups, the most basic cases are cyclic groups and permutation groups, and Cay-ley’s Theorem tells us that for any finite group G, there exists n ≤ |G|, such that G is asubgroup of Symn. Burnside wrote a book on finite groups, bringing group theory intoa new era. Associated with Lie groups (a geometry object), Coxeter groups became animportant topic in the last century. These groups have beautiful representations throughreflections in Euclidean space. Mathematicians generalized mirrors of these reflectionsto root systems for more general study. Since then, some mathematicians transferredtheir attention to these groups and their root systems to define and study many algebras,such as Birman-Murakami-Wenzl algebras, Hecke algebras, Temperley-Lieb algebras, andBrauer algebras (our main topic in this thesis).In this thesis, our main task is to complete the definition and explore basic properties ofthe Brauer algebras associated to finite (spherical) Coxeter groups.

0.2 The importance of Brauer algebras

Brauer algebras were first defined by Brauer in [4] for studying the invariant theory of or-thogonal matrix groups. It has a closely related algebra, the Birman-Murakami-Wenzl al-gebra. Birman and Wenzl [3] and Murakami [34], independently reversed Jones’ processto introduce algebras, commonly known as Birman-Murakami-Wenzl (BMW in short), asan algebraic framework for the Kauffman polynomials. The Brauer algebra is the limitform of the BMW algebra of the same type (evaluating the parameters at special values).The Brauer algebras and BMW algebras are related to Artin groups of type A, Iwahori-Hecke algebras of type A and Temperley-Lieb algebras. All algebras mentioned so farfeature in a vast variety of mathematical subjects, including combinatorial and geometricrepresentation theory, knot theory, homological algebra, the study of subfactors, topo-

2 INTRODUCTION

logical quantum field theory and quantum groups. Moreover, they are also examples ofcellular algebras, in the sense of Graham and Lehrer [26], whose theory leads to infor-mation about their representation theory and a semisimplicity criterion for the algebras.Furthermore, they are connected to diagram algebras, which originated from importantproblems in statistical mechanics and whose representation theory has not only helpedsolve these problems but also often led to further questions and fruitful interplay. TheTemperley-Lieb algebra in the loop representation with zero loop factor coincides withan exactly solvable model of two-dimensional critical dense polymers which is a model ofmuch interest in physics since it gives rise, in the continuum scaling limit, to a logarithmicconformal field theory (see [38]).

0.3 Diagram algebras

In [32], Morton and Wasserman proved that the BMW algebra is isomorphic to the Kauff-man tangle algebra, an algebra of (regular isotopy equivalence class of) tangles on nstrands in the solid cylinder modulo the Kauffman skein relation (see [28] and [31]).Furthermore, the BMW algebras may be constructed as a deformation of the Braueralgebras obtained by replacing the symmetric group algebras with the correspondingIwahori-Hecke algebras. By no longer distinguishing between over and under crossingof strands occurring in the tangle diagrams, one obtains the Brauer algebra as a classicallimit of the BMW algebra. All of these algebras are diagram algebras; that is, algebraswith basis a given set of diagrams where multiplication is described by a simple diagramcalculus which is both intuitive and computationally effective. Moreover, having a geo-metric realization allows one to see certain properties of these algebras very clearly, likethe existence of Markov traces, conditional expectations and good canonical bases.

0.4 The aims and results of the thesis

In view of these relations between the Brauer algebras and several objects of type A,it is natural to seek analogues of these algebras for other types of spherical Coxetergroup. Generalizations for other simply laced types (type D and E) of the Temperley-Liebalgebras were first introduced by Graham [25] and Fan [22] and of Brauer algebras andBMW algebras and BMW algebra by Cohen, Frenk, Gijsbers, and Wales [9], [10], [12],[13], [17].It is reasonable to ask how to define the Brauer algebras of non-simply laced types. Tits[39] wrote about how to get Coxeter groups of non-simply laced type from simply lacedones as subgroups of invariant elements of non-trivial isomorphisms on Dynkin diagrams;Mühlherr [33] showed how to get Coxeter groups as subgroups of Coxeter groups byadmissible partitions of the canonical set of generators. Both of these two works inspiredus to construct the Brauer algebras of non-simply laced types from simply laced ones.In this thesis, our main aim is to get the spherical non-simply laced Brauer algebrasfrom simply laced types by use of the canonical symmetry or partitions defined on theirDynkin diagrams as in [39] and [33]( also see [6], [7], [36], [37]). Their cellularity isalso discussed. Our conclusion is summarized in the following table. Here BSO=beingsubalgebra of. Especially, it is proved that the Brauer algebra of type Cn is a stratified

0.4 THE AIMS AND RESULTS OF THE THESIS 3

Table 1: Main results

type rank BSO cellularity

Cn

∑ni=0

�

∑

p+2q=in!

p!q!(n−i)!

�22n−i (n− i)! A2n−1 cellular

Bn 2n+1 · n!!− 2n · n!+ (n+ 1)!!− (n+ 1)! Dn+1 cellularF4 14985 E6 stratified cellularH3 1045 D6 stratified cellularH4 236025 E8 stratified cellular

In2 (n≥ 5) 2n+ n2(n odd), 2n+ 3

2n2(n even) An−1 stratified cellular

G2 39 D4 stratified cellular

cellular algebra as defined in [2].One may pose the following two questions.

(i) If X , Y are Weyl types (Definition 1.1.7) such that X is a subdiagram of Y , is Br(X )a subalgebra of Br(Y )?

(ii) Is the Temperley-Lieb algebra defined by Graham in [25] always the same as thesubalgebra defined by the ei?

For the first question, we give a positive answer in Proposition 1.6.7. For the second, theanswer is a little disappointing, but we find that our Temperley-Lieb algebras of type Ccoincide with the Temperley-Lieb algebras of type B introduced in [20] by means diagrampresentations.

4 INTRODUCTION

1PRELIMINARY KNOWLEDGE

Brauer algebras were first studied by Brauer [4] for the Weyl Duality Theorem. In thisthesis, Brauer algebras associated to spherical Dynkin diagrams will be studied. This the-sis introduces our recent results about Brauer algebras of non-simply laced type and alsorecalls some consequences of the definition and the canonical basis of Brauer algebras ofsimply laced type.

At the beginning of this chapter, we first recall necessary classical knowledge aboutCoxeter groups, Coxeter diagrams, their root systems and a few related algebras suchas Hecke algebra and Temperley-Lieb algebra. We also introduce several recent resultsabout a partial order on some mutually orthogonal root sets associated to Coxeter groupsof simply laced type, and also the corresponding Brauer algebras. Subsequently we in-troduce the classical Brauer algebra and prove some similar results to those in [17]. InSection 1.6, we introduce the definition of Brauer algebras of type Bn, Cn, F4, and G2,and some of their basic properties. At the end, we state the definition of BMW algebrasof simply laced types and the difficulty of obtaining BMW algebras of non-simply lacedtype.

1.1 Coxeter groups and Weyl groups

This section is based on the book [1].

Definition 1.1.1. Let I be a set. A Coxeter matrix over I is a matrix M = (mi j)i, j∈I wheremi j ∈ N∪ {∞} with mii = 1 for i ∈ I , and mi j = m ji > 1 for distinct i, j ∈ I . The Coxetergroup W (M), or just W , of type M is the group with presentation

¬

{ri | i ∈ I} | (ri r j)mi j = 1

¶

.

This means that W is freely generated by the set S = {ri | i ∈ I} subject to the relations(ri r j)mi j if mi j ∈ N and no such relation if mi j = ∞. The pair (W, S) is called a Coxetersystem of type M .

Definition 1.1.2. The Coxeter matrix M is often described by a labeled graph Γ(M)whose vertex set is I and in which two nodes i and j are joined by an edge labeled mi j ifmi j > 2. If mi j = 3, then the label 3 of the edge {i, j} is often omitted. If mi j = 4, then

6 PRELIMINARY KNOWLEDGE

instead of the label 4 at the edge {i, j} one often draws a double bond. If mi j = 6, theninstead of the label 6 at the edge {i, j} one often draws a triple bond. This labeled graphis called the Coxeter diagram of M .

Definition 1.1.3. When referring to a connected component of a Coxeter matrix M overI , we view M as a labeled graph. In other words, a connected component of M is amaximal connected subset J of I such that m jk = 2 for each j ∈ J and k ∈ I \ J . If M hasa single connected component, it is called connected or irreducible. A Coxeter group Wover a Coxeter diagram is called irreducible if M is irreducible.

Definition 1.1.4. Let X and Y be free Z-modules of rank n with a Z-bilinear form ⟨·, ·⟩ :X × Y → Z. Let Φ be a finite subset of X and suppose that for each α ∈ Φ we have acorresponding element α∨ in Y . Set Φ∨ = {α∨ | α ∈ Φ}. Given α ∈ Φ, we define the linearmap sα : X → X by

sα(x) = x −

x ,α∨�

α

and similarly the linear map s∨α : Y → Y by

s∨α(y) = y −

α, y�

α∨.

Definition 1.1.5. Let (X , Y,Φ,Φ∨) be a quadruple as above. We say (X , Y,Φ,Φ∨) is a rootdatum if the following three conditions are satisfied for every root α ∈ Φ.

(i) sα and s∨α are reflections.

(ii) Φ is closed under the action of sα.

(iii) Φ∨ is closed under the action of s∨α .

We call elements of the finite set Φ roots and the elements of Φ∨ coroots. The groupgenerated by all sα is called the Weyl group of the root datum.A subset Π of Φ is called a root base if

(i) Π is a basis of R-span of Φ,

(ii) each root β ∈ Φ can be written as β =∑

α∈Π kαα with integer coefficients kα allnonnegative or all nonpositive.

Further the sum of all |kα| for β is called the height of β .

Definition 1.1.6. If Π is a root base of roots Φ, then the root lengths of the simple rootsare often registered in the Coxeter diagram by adding an arrow on the labeled edgebetween the nodes i and j pointing towards j if the length of the root belonging to i isbigger than the length belonging to j. The resulting diagrams is known as the Dynkindiagram.

Definition 1.1.7. A Coxeter diagram is said to be of Weyl type if it only has connectedcomponents of type An (n ≥ 1), Bn (n ≥ 2), Cn (n ≥ 3), Dn (n ≥ 4), En (6 ≤ n ≤ 8), F4,G2 as in Table 1.1.

1.1 COXETER GROUPS AND WEYL GROUPS 7

Table 1.1: Coxeter diagrams of spherical types

name diagramAn ◦

n◦

n−1◦

n−2· · · · · · ◦

2◦1

Dn ◦n

◦n−1· · · · · · ◦

4

2◦

◦3

◦1

En, 6≤ n≤ 8 ◦n

◦n−1· · · · · · ◦

5

2◦

◦4

◦3

◦1

Bn ◦n−1

◦n−2· · · · · · ◦

2◦1> ◦

0Cn ◦

n−1◦

n−2· · · · · · ◦

2◦1< ◦

0F4 ◦

1◦2< ◦

3◦4

H3 ◦1

5 ◦2

◦3

H4 ◦1

5 ◦2

◦3

◦4

In2 ◦

0

n ◦1

G2 ◦0< ◦

1

It is known that the irreducible finite (spherical) Coxeter groups are those of type An,Bn(Cn), Dn, En (6 ≤ n ≤ 8), F4, G2, H3, H4, and In

2. We list their Coxeter diagrams inTable 1.1. Let V = RI be the Euclidean space of dimension l = |I | with basis {αi}i∈Π. LetB be the bilinear form over V such that

B(αi ,α j) =−2 cosπ

mi j.

Let ρi(x) = x − B(x ,αi)αi , for x ∈ V , i ∈ I . Then the map ρ : I → GL(V ), ρ(ri) = ρi candetermine a faithful representation of W (M) on RI ([1] or [6]).

Definition 1.1.8. The subset Φ = ∪i∈Iρ(W )αi of V is called the root system of W . Thesubset of Φ+ ⊂ Φ (Φ− ⊂ Φ ) of all elements with nonnegative (nonpositive) coefficientsfor the basis {αi}i∈I is called the set of positive roots (negative roots) of W . Each root αiis called a simple root of W for i ∈ I .

It is known from [1, Theorem 3, Chapter 6] that

Φ = Φ+ ∪Φ−, Φ+ =−Φ−, Φ+ ∩Φ− = ;.


1.2 A poset of simply laced type

Let Q be a spherical Coxeter diagram of simply laced type, i.e., its connected componentsare of type A, D, E as listed in Table 1.1.

When Q is An, Dn, E6, E7, or E8, we denote it as Q ∈ ADE. Let (W, T ) be the Coxetersystem of type Q with T = {R1, . . . , Rn} associated to the diagram of Q in Table 1.1. Let Φbe the root system of type Q, let Φ+ be its positive root system, and let αi be the simpleroot associated to the node i of Q. We are interested in sets B of mutually commutingreflections, which has a bijective correspondence with sets of mutually orthogonal rootsof Φ+, since each reflection in W is uniquely determined by a positive root and vice versa.

Remark 1.2.1. The action of w ∈W on B is given by conjugation in case B is described byreflections and given by w{β1, . . . ,βp} = Φ+ ∩ {±wβ1, . . . ,±wβp}, in case B is describedby positive roots. For example, R4R1R2R1{α1 +α2,α4}= {α1 +α2,α4}, where Q = A4.

For α, β ∈ Φ, we write α∼ β to denote |(α,β)|= 1. Thus, for i and j nodes of Q, wehave αi ∼ α j if and only if i ∼ j.

Definition 1.2.2. Let B be a W -orbit of sets of mutually orthogonal positive roots. Wesay that B is an admissible orbit if for each B ∈ B, and i, j ∈ Q with i 6∼ j and γ,γ−αi+α j ∈ B we have riB = r jB, and each element in B is called an admissible root set.

This is the definition from [11], and there is another equivalent definition in [9]. Wealso state it here.

Definition 1.2.3. Let B ⊂ Φ+ be a mutually orthogonal root set. If for all γ1, γ2, γ3 ∈ Band γ ∈ Φ+, with (γ,γi) = 1, for i = 1, 2, 3, we have 2γ+ γ1 + γ2 + γ3 ∈ B, then B iscalled an admissible root set.

By these two definitions, it follows that the intersection of two admissible root setsare admissible. It can be checked by definition that the intersection of two admissiblesets are still admissible. Hence for a given set X of mutually orthogonal positive roots,the unique smallest admissible set containing X is called the admissible closure of X , anddenoted as X cl (or X ). Up to the action of the corresponding Weyl groups, all admissibleroot sets of type An, Dn, E6, E7, E8 have appeared in [9], [13] and [17], and are listedin Table 1.2. In the table, the set Y (t)∗ consists of all α∗ for α ∈ Y (t), where α∗ is theunique positive root orthogonal to α and all other positive roots orthogonal to α for typeDn with n> 4.

Example 1.2.4. If Q = D4, the root set {α1,α2,α4} is mutually orthogonal but not admis-sible, and its admissible closure is {α1,α2,α4,α1 +α2 + 2α3 +α4}.

Definition 1.2.5. LetA denote the collection of all admissible subsets of Φ consisting ofmutually orthogonal positive roots. Members ofA are called admissible sets.

Now we consider the actions of Ri on an admissible W -orbit B. When RiB 6= B, Wesay that Ri lowers B if there is a root β ∈ B of minimal height among those moved by Rithat satisfies β − αi ∈ Φ+ or RiB < B. We say that Ri raises B if there is a root β ∈ B ofminimal height among those moved by Ri that satisfies β + αi ∈ Φ+ or RiB > B. By thiswe can set an partial order on B =W B. The poset (B,<) with this minimal ordering iscalled the monoidal poset (with respect to W ) on B (so B should be admissible for theposet to be monoidal). If B just consists of sets of a single root, the order is determined

1.3 BRAUER ALGEBRAS OF SIMPLY LACED TYPES 9

Table 1.2: Admissible root sets of simply laced type

Q representatives of orbits under W (Q)An {α2i−1}ti=1, 0≤ t ≤ b(n+ 1)/2c .Dn Y (t) = {αn+2−2i ,αn−2, . . . ,αn+2−2t} 0≤ t ≤ bn/2c .

{αn+2−2i ,αn−2, . . . ,α4,α1} if 2|nY (t)∪ Y (t)∗ 0≤ t ≤ bn/2c

E6 ;, {α6}, {α6,α4}, {α6,α2,α3}cl

E7 ;, {α7}, {α7,α5}, {α5,α5,α2}, {α7,α2,α3}cl, {α7,α5,α2,α3}cl

E8 ;, {α8}, {α8,α6}, {α8,α2,α3}cl, {α8,α5,α2,α3}cl

by the canonical height function on roots. There is an important conclusion in [11],stated below. This theorem plays a crucial role in obtaining a basis for Brauer algebra ofsimply laced type in [9].

Theorem 1.2.6. There is a unique maximal element in B.

1.3 Brauer algebras of simply laced types

The Brauer algebra of type A was first introduced in [4] for studying the invariant theoryof orthogonal group. In [9], it is extended to simply laced types, in the way describedbelow.

Definition 1.3.1. Let R be a commutative ring with invertible element δ and let Q be asimply laced Coxeter diagram. The Brauer algebra of type Q over R with loop parameter δ,denoted Br(Q, R,δ), is the R-algebra generated by Ri and Ei , for each node i of Q subjectto the following relations, where ∼ denotes adjacency between nodes of Q.

R2i = 1 (1.3.1)

E2i = δEi (1.3.2)

Ri Ei = EiRi = Ei (1.3.3)

RiR j = R jRi , for i� j (1.3.4)

EiR j = R j Ei , for i� j (1.3.5)

Ei E j = E j Ei , for i� j (1.3.6)

RiR jRi = R jRiR j , for i∼ j (1.3.7)

R jRi E j = Ei E j , for i∼ j (1.3.8)

Ri E jRi = R j EiR j , for i∼ j (1.3.9)

As before, we call Br(Q) := Br(Q,Z[δ±1],δ) the Brauer algebra of type Q and denote byBrM(Q) the submonoid of the multiplicative monoid of Br(Q) generated by δ±1 and allRi and Ei .

For any β ∈ Φ+ and i ∈ {1, . . . , n}, there exists a w ∈ W such that β = wαi . ThenRβ := wRiw

−1 and Eβ := wEiw−1 are well defined (this is well known from Coxeter


group theory for Rβ ; see [9, Lemma 4.2] for Eβ). If β ,γ ∈ Φ+ are mutually orthogonal,then Eβ and Eγ commute (see [9, Lemma 4.3]). Hence, for B ∈A , we define the product

EB =∏

β∈B

Eβ , (1.3.10)

which is a quasi-idempotent, and the normalized version

EB = δ−|B|EB, (1.3.11)

which is an idempotent element of the Brauer monoid. For a mutually orthogonal rootsubset X ⊂ Φ+, we have

EX cl = δ|Xcl\X |EX . (1.3.12)

Let CX = {i ∈ Q | αi ⊥ X } and let W (CX ) be the subgroup generated by the generatorsof nodes in CX . The subgroup W (CX ) is called the centralizer of X . The normalizer of X ,denoted by NX can be defined as

NX = {w ∈W | EX w = wEX }.

We let DX denote a set of right coset representatives for NX in W .In [9], an action of the Brauer monoid BrM(Q) on the collection A of admissible rootsets in Φ+ was indicated below, where Q ∈ ADE.

Definition 1.3.2. There is an action of the Brauer monoid BrM(Q) on the collection A .The generators Ri (i = 1, . . . , n) act by the natural action of Coxeter group elements onits positive root sets as in Remark 1.2.1, and the element δ acts as the identity, and theaction of Ei (i = 1, . . . , n) is defined by

EiB :=

B if αi ∈ B,

(B ∪ {αi})cl if αi ⊥ B,

RβRiB if β ∈ B \α⊥i .

(1.3.13)

We will refer to this action as the admissible set action. This monoid action plays animportant role in getting a basis of BrM(Q) in [9]. For the basis, we state one conclusionfrom [9] below.

Proposition 1.3.3. Each element of the Brauer monoid BrM(Q) can be written in the form

δkuEX zv,

where X is the highest element from one W-orbit inA , u, v−1 ∈ DX , z ∈W (CX ), and k ∈ Z.

1.4 Cellular algebra and cellularly stratified algebra

This section is an introduction to cellular algebras and cellularly stratified algebras. Weprove that each of our Brauer algebras over a special ground ring belongs to either ofthem in the next chapters, which might help us to understand their structures and theirrepresentations.Recall from [26] that an associative algebra A over a commutative ring R is cellular ifthere is a quadruple (Λ, T, C ,∗) satisfying the following three conditions.

1.4 CELLULAR ALGEBRA AND CELLULARLY STRATIFIED ALGEBRA 11

(C1) Λ is a finite partially ordered set. Associated to each λ ∈ Λ, there is a finite setT (λ). Also, C is an injective map

∐

λ∈Λ

T (λ)× T (λ)→ A

whose image is an R-basis of A.

(C2) The map ∗ : A → A is an R-linear anti-involution such that C(x , y)∗ = C(y, x)whenever x , y ∈ T (λ) for some λ ∈ Λ.

(C3) If λ ∈ Λ and x , y ∈ T (λ), then, for any element a ∈ A,

aC(x , y)≡∑

u∈T (λ)

ra(u, x)C(u, y) mod A<λ,

where ra(u, x) ∈ R is independent of y and where A<λ is the R-submodule of Aspanned by {C(x ′, y ′) | x ′, y ′ ∈ T (µ) for µ < λ}.

Such a quadruple (Λ, T, C ,∗) is called a cell datum for A.There is also an equivalent definition due to König and Xi.

Definition 1.4.1. Let A be an R-algebra. Assume that there is an anti-automorphism ion A with i2 = id. A two sided ideal J in A is called cellular if and only if i(J) = J ,there exists a left ideal ∆⊂ J such that ∆ has finite rank and there is an isomorphism ofA-bimodules α : J '∆⊗R i(∆) making the following diagram commutative:

J α //

i

��

∆⊗R i(∆)

x⊗y→i(y)⊗i(x)��

J α // ∆⊗R i(∆)

The algebra A is called cellular if there is a vector space decomposition A= J ′1 ⊕ · · · ⊕ J ′nwith i(J ′j) = J ′j for each j such that setting J j = ⊕

jk=1J ′j gives a chain of two sided ideals

of A with the property for each j the quotient J ′j = J j/J j−1 is a cellular ideal of A/J j−1.

We also recall the definitions of iterated inflations from [29] and cellularly stratifiedalgebras from [27]. Given an R-algebra B, a finitely generated free R-module V , and abilinear form ϕ : V⊗RV −→ B with values in B, we define an associative algebra (possiblywithout unit) A(B, V,ϕ) as follows: as an R-module, A(B, V,ϕ) equals V ⊗R V ⊗R B. Themultiplication is defined on basis element as follows:

(a⊗ b⊗ x)(c⊗ d ⊗ y) := a⊗ d ⊗ xϕ(b, c)y.

Assume that there is an involution i on B. Assume, moreover, that i(ϕ(v, w)) = ϕ(w, v).We can extend this involution i to A(B, V,ϕ) by defining i(a⊗ b⊗ x) = b⊗ a⊗ i(x); thenwe call A(B, V,ϕ) an inflation of B along V . Let B be an inflated algebra (possibly withoutunit) and C be an algebra with unit. We define an algebra structure in such a way thatB is a two-sided ideal and A/B = C . We require that B is an ideal, the multiplicationis associative, and that there exists a unit element of A which maps onto the unit of thequotient C . We call A an inflation of C along B, or iterated inflation of C along B. Wepresent Proposition 3.5 and Theorem 4.1 of [29].


Proposition 1.4.2. An inflation of a cellular algebra is cellular again. In particular, aniterated inflation of n copies of R is cellular, with a cell chain of length n as in Definition1.4.1.

More precisely, the second statement has the following meaning. Start with C a fullmatrix ring over R and B an inflation of R along a free R-module, and from a new A whichis an inflation of the old A along the new B, and continue this operation. Then after nsteps we have produced a cellular algebra A with a cell chain of length n.

Theorem 1.4.3. Any cellular algebra over R is the iterated inflation of finitely many copiesof R. Conversely, any iterated inflation of finitely many copies of R is cellular.

Let A be cellular (with identity) which can be realized as an iterated inflation ofcellular algebras Bl along vector spaces Vl for l = 1, . . . , n. This implies that as a vectorspace

A=⊕nl=1Vl ⊗ V1 ⊗ Bl ,

and A is cellular with a chain of two sided ideals 0 = J0 ⊂ J1 · · · ⊂ Jn = A, which can berefined to a cell chain, and each quotient Jl/Jl−1 equals Vl⊗V1⊗Bl as an algebra withoutunit. The involution i of A is defined through the involution il of the algebra Bl wherei(a⊗ b⊗ x) = b⊗ a⊗ jl(x). The multiplication rule of a layer Vl ⊕V1⊕Bl is indicated by

(a⊗ b⊗ x)(c⊗ d ⊗ y) := a⊗ d ⊗ xϕ(b, c)y + lower terms.

Here lower terms refers to elements in lower layers Vh⊗ Vh⊗ Bh for h< l. Let 1Blbe the

identity of the algebra Bl .

Definition 1.4.4. Let R be a field. A finite dimensional associative algebra A is calledcellularly stratified with stratification data (B1, V1, . . . , Bn, Vn) if and only if the followingconditions are satisfied:

(1) The algebra is an iterated inflation of cellular algebra Bl along vector spaces Vl forl = 1,. . ., n.

(2) For each l = 1,. . ., n, there exist ul , vl such that el = ul ⊗ vl ⊗ 1Blis an idempotent.

(3) If l > m, then el em = em = emel .

1.5 The classical Brauer algebra

Let m ∈ N. In this section, we describe the root system of the Coxeter group of typeAm, focusing on collections of mutually orthogonal positive roots called admissible sets(Definition 1.2.5). Also, the notion of height for elements of the Brauer monoid BrM(Am)is introduced and discussed. A major goal, established in Theorem 1.5.10 from [14], isto exhibit a normal form for elements of the monoid BrM(Am) as a product of generators.

1.5 THE CLASSICAL BRAUER ALGEBRA 13

Remark 1.5.1. As a consequence of the relations of Brauer algebras Br(Q) of simply lacedtype, it is straightforward to show that the following relations hold in Br(Q) for all nodesi, j, k with i ∼ j ∼ k and i 6∼ k (see [9, Lemma 3.1]).

EiR jR j = Ei E j (1.5.1)

R j Ei E j = Ri E j (1.5.2)

EiR j Ei = Ei (1.5.3)

E j EiR j = E jRi (1.5.4)

Ei E j Ei = Ei (1.5.5)

E j EiRk E j = E jRi Ek E j (1.5.6)

E jRiRk E j = E j Ei Ek E j (1.5.7)

Remark 1.5.2. In [4], Brauer gives a diagrammatic description for a basis of the Braueralgebra of type Am. Each basis element is a diagram with 2m+2 dots and m+1 strands,where each dot is connected by a unique strand to another dot. Here we suppose the2m+2 dots have coordinates (i, 0) and (i, 1) in R2 with 1≤ i ≤ m+1. The multiplicationof two diagrams is given by concatenation, where any closed loops formed are replacedby a factor of δ. The generators Ri and Ei of Br(Am) correspond to the diagrams indicatedin Figure 1.1.

. . . . . .

1 i−1 i i+1 i+2 m+1

Ri

. . . . . .

1 i−1 i i+1 i+2 m+1

Ei

Figure 1.1: Brauer diagrams corresponding to Ri and Ei .

Each Brauer diagram can be written as a product of elements from {Ri , Ei}mi=1. Thisstatement is illustrated in Figure 1.2.

Henceforth, we identify BrM(Am) with its diagrammatic version. It makes clear thatBr(Am) is a free algebra over Z[δ±1] of rank (m + 1)!!, the product of the first m + 1positive odd integers. The monomials of BrM(Am) that correspond to diagrams will bereferred to as diagrams.

Definition 1.5.3. Let m ≥ 1. The root system of the Coxeter group W (Am) of type Am isdenoted by Φ. It is realized as Φ := {εi − ε j | 1 ≤ i, j ≤ m+ 1, i 6= j} in the Euclideanspace Rm+1, where εi is the ith standard basis vector. Put αi := εi − εi+1. Then {αi}mi=1is called the set of simple roots of Φ. Denote by Φ+ the set of positive roots in Φ withrespect to these simple roots; that is, Φ+ := {εi − ε j | 1≤ i < j ≤ m+ 1}.

We review some facts about A by diagrams. An admissible set B corresponds to aBrauer diagram top in the following way: for each β ∈ B, where β = εi−ε j for some i, j ∈{1, . . . , m+ 1} with i < j, draw a horizontal strand in the corresponding Brauer diagram


V = V

Figure 1.2: A Brauer diagram and a visualization of it as the productR2R5E1R3R6E2E4V E3E5E7R2E4E6R1E3E5R2E4, where V can be either the identity or thesimple crossing R1.

top from the dot (i, 1) to the dot ( j, 1). All horizontal strands on the top are obtainedthis way, so there are precisely |B| horizontal strands. The top of the Brauer diagram ofFigure 1.2, corresponds to the admissible set {α1+α2,α2+α3+α4+α5,α5+α6+α7}.

Similarly, there is an admissible set corresponding to a Brauer diagram bottom. Thebottom of the Brauer diagram of Figure 1.2, corresponds to the admissible set {α1+α2+α3 +α4 +α5 +α6 +α7,α2 +α3 +α4 +α5,α4}.

Alternatively, this action can be described as follows for a monomial a: completethe top corresponding to B into a Brauer diagram b, without increasing the number ofhorizontal strands in the top. Now aB is the top of the Brauer diagram ab. We will makeuse of this action in order to provide a normal form for elements of BrM(Am). In [9,Definition 3.2], it is shown that this action is well defined for any spherical simply lacedtype.

For the admissible set action, there is a right action of BrM(Am) on A defined byX a = aopX for a ∈ BrM(Am) and X ∈ A . In order to interpret the right action of a onB, the latter should be pictured as the bottom of a Brauer diagram, so that the bottomcorresponding to Ba is the bottom of ba. Observe that a; is the top of the Brauer diagramof a (i.e., the collection of top horizontal strands of a and top row of points) and ;a is itsbottom (i.e., the collection of bottom horizontal strands of a and bottom row of points).

Recall that the height ht(β) of a positive root β ∈ Φ+ is h if it is the sum of preciselyh simple roots. This definition will be extended to a height function onA in such a waythat ht({β}) = ht(β)− 1.

Definition 1.5.4. The height of an admissible set B, denoted by ht(B), is the minimalnumber of crossings in a completion of the top corresponding to B to a Brauer diagramwithout increasing the number of horizontal strands at the top.

For example, the height of the top of the Brauer diagram of Figure 1.2 is equal to 4

1.5 THE CLASSICAL BRAUER ALGEBRA 15

and the height of the bottom is equal to 3.

Definition 1.5.5. For every element a ∈ BrM(Am), we define the height of a, denotedby ht(a), as the minimal number of generators Ri needed to write a as a product of thegenerators R1, . . . , Rm, E1, . . . , Em, δ, δ−1.

In terms of Brauer diagrams, the height of a is the minimal number of crossingsneeded to draw a. Consequently, the height of an admissible set B is the minimal heightover all possible Brauer diagram completions of B.

The Brauer diagram of Figure 1.2 has height 7 if V is the identity and height 8 ifV = R1. The lemma below states some useful properties of this height function.

Remark 1.5.6. There is a natural anti-involution on Br(Am), denoted by x 7→ xop, deter-mined by

Ri 7→ Ri and Ei 7→ Ei .

By anti-involution, we mean a Z[δ±1]-linear anti-automorphism whose square is theidentity.

Lemma 1.5.7. Let B, C ∈A with |B|= |C |. Then there is a unique diagram aB,C of heightht(B) + ht(C) in BrM(Am) such that aB,C; = B and ;aB,C = C. This diagram satisfiesaB,C aop

B,C = δ|B|EB as well as aB,C C = B and BaB,C = C.

Proof. The easiest proof to our knowledge is based on diagrams.There is a unique way to complete a given top and bottom to a Brauer diagram with

a minimal number of crossings: connect the first dot at the top from the left that is notthe endpoint of a horizontal strand at the top to the first dot at the bottom that is not theendpoint of a horizontal strand at the bottom; proceed similarly with the second, and soon, until the Brauer diagram is complete. If the top corresponds to B and the bottom toC , the resulting diagram is the required monomial aB,C .

Lemma 1.5.8. Suppose B ∈ A has height 0. Then there are r = m− 2|B| diagrams ofheight 1 in the group of invertible elements in EBBrM(Am)EB forming a Coxeter system oftype Ar . (Here, invertibility is meant with respect to the unit EB of the monoid).

Proof. As discussed above, a Brauer diagram with top and bottom corresponding to B hasr + 1 free dots at the top and also r + 1 at the bottom. For a diagram to be an invertibleelement in EBBrM(Am)EB, the remaining strands need to be vertical, so they belong tothe symmetric group on the r + 1 free dots at the top (or those at the bottom). Now,up to powers of δ, the idempotent EB is the element in which all r vertical strands donot cross. Selecting diagrams in which the ith and (i + 1)st vertical strands cross and noothers (for i = 1, 2, . . . , r), we find the required Coxeter system of type Ar .

Definition 1.5.9. For B ∈A of height 0, denote by KB the Coxeter group determined byLemma 1.5.8.

In the middle part of the right hand side of Figure 1.2, next to V , the element e3e5e7 =EB appears, where B = {α3,α5,α7} has height 0. Now KB is a Coxeter group of type A1,generated by R1 EB, so the choices for V are consistent with the possibilities for V EB =EBV EB ∈ KB.

The following result is known; see for instance [18, Lemma 2.2].


Theorem 1.5.10. Let i ∈ {0, 1, . . . , bm/2c} and let B be any admissible set of size i and ofheight 0. Then each element a of BrM(Am) with |a;|= i can be written uniquely as

δkUVW

for certain k ∈ Z, U a diagram in BrM(Am)EB with UB = a;, W a diagram in EBBrM(Am)with ;a = BW, and V ∈ KB such that

ht(a) = ht(U) + ht(V ) + ht(W ).

Proof. Take U = aa;,B and W = aop;a,B. Then V = UopaW op has top and bottom equal to B

and so belongs to KB. By Lemma 1.5.7 and (1.3.11),

UVW = aa;,Baopa;,Baa;a,Baop

;a,B = δ4i Ea;aE;a = δ

ka

for k = 4i. As ht(U) = ht(a;) and ht(W ) = ht(;a) and ht(V ) is the length of V withrespect to the Coxeter system of Lemma 1.5.8, this proves that a has a decomposition asstated.

As for uniqueness, suppose a = UVW is a product decomposition as stated. Thena; = UV B = UB = U; (as U = U EB) and ;U = B, so by Lemma 1.5.8 and minimalityof the height of U , we find U = aa;,B. Similarly, W can be shown to be equal to aop

;a,B.

Finally, V = EBV EB = δ−4|B|UopUVWW op = δ−4|B|UopaW op is uniquely determined by a,U , and W .

There is a more general version for simply laced types in [17]. We keep notation asin [17, Section 2] and first introduce some basic concepts. Let M be the diagram of aconnected finite simply laced Coxeter group (type An, Dn, E6, E7, E8). BrM(M) is theassociated Brauer monoid as Definition 1.3.1. By BY we denote the admissible closure of{αi |i ∈ Y }, where Y is a coclique of M . The set BY is a minimal element in the W (M)-orbit of BY which is endowed with a poset structure induced by the partial ordering <defined on W (M)-orbits in A in Section 1.2. If d is the Hasse diagram distance forW (M)BY from BY to the unique maximal element, then for B ∈ W (M)BY the height ofB, already used in Definition notation ht(B), is d− l, where l is the distance in the Hassediagram from B to the maximal element. The Figure 1.3 is a Hasse diagram of admissiblesets of type A4 with 2 mutually orthogonal positive roots. As indicated in Theorem 1.2.6,the set {α1 +α2 +α3,α2 +α3 +α4} is the maximal root set in its W (A4)-orbit.

Theorem 1.5.11. ([17, Theorem 2.7]) Each monomial a in BrM(M) can be uniquely writ-ten as δiaB EY haop

B′ for some i ∈ Z and h ∈W (MY ), where W (MY ) is the group of invertibleelements in EY W (M)EY , B = a;, B

′= ;a, aB ∈ BrM(E6), aop

B′ ∈ BrM(E6) and(i) a;= aB;= aBBY , ;a = ;aop

B′ = BY aopB′ ,

(ii) ht(B) =ht(aB), ht(B′) =ht(aopB′ ).

1.6 DEFINITION OF BRAUER ALGEBRAS OF TYPE Bn, Cn, F4, G2 17

{α1, α3} {α1 + α2 + α3, α2} {α1, α4} {α2, α4} {α2 + α3 + α4, α3}

{α1 + α2, α2 + α3} {α1, α3 + α4} {α1 + α2 + α3 + α4, α2} {α1 + α2, α4} {α2 + α3, α3 + α4} {α1 + α2 + α3 + α4, α3}

{α1 + α2, α2 + α3 + α4} {α1 + α2 + α3, α3 + α4} {α1 + α2 + α3 + α4, α2 + α3}

{α1 + α2 + α3, α2 + α3 + α4}

Figure 1.3: A Hasse diagram of type A4.

1.6 Definition of Brauer algebras of type Bn, Cn, F4, G2

We denote M ∈ BCFG, if M is a Dynkin diagram of type Bn, Cn, F4 or G2. We abuse thenotation i ∈ M for a node i of M .

Definition 1.6.1. Let R be a commutative ring with invertible element δ. For n ∈ N, theBrauer algebra of type M ∈ BCFG over R with loop parameter δ, denoted by Br(M , R,δ), isthe R-algebra generated by {ri , ei}i∈M subject to the following relations. For each i ∈ M ,

r2i = 1, (1.6.1)

riei = ei ri = ei , (1.6.2)

e2i = δκi ei; (1.6.3)

for i, j ∈ M not adjacent to each other, namely ◦i◦j,

ri r j = r j ri , (1.6.4)

ei r j = r jei , (1.6.5)

eie j = e jei; (1.6.6)


for i, j ∈ M and ◦i

◦j,

ri r j ri = r j ri r j , (1.6.7)

r j rie j = eie j , (1.6.8)

rie j ri = r jei r j; (1.6.9)

for i, j ∈ M and ◦i> ◦

j,

r j ri r j ri = ri r j ri r j , (1.6.10)

r j rie j = rie j , (1.6.11)

r jei r jei = eie jei , (1.6.12)

(r j ri r j)ei = ei(r j ri r j), (1.6.13)

e j rie j = δe j , (1.6.14)

e jeie j = δe j , (1.6.15)

e j ri r j = e j ri , (1.6.16)

e jei r j = e jei; (1.6.17)

for i, j ∈ M and ◦i< ◦

j,

rie jei = r jei , (1.6.18)

eie j ri = ei r j , (1.6.19)

e j rie j rie j = e j , (1.6.20)

e j rie j ri r j = e j ri r j ri , (1.6.21)

ei r jei = δ2ei (1.6.22)

r j rie j rie j = ri r j rie j . (1.6.23)

(r j ri)6 = 1. (1.6.24)

The parameter κi ∈ N is given below,for type Cn, κ0 = 1, κi = 2 for 1≤ i ≤ n− 1;for type Bn, κ0 = 2, κi = 1 for 1≤ i ≤ n− 1;for type F4, κ1 = κ2 = 2, κ3 = κ4 = 1;for type G2, κ0 = 3, κ1 = 1.If R = Z[δ±1] we write Br(M) instead of Br(M , R,δ) and speak of the Brauer algebra oftype M . The submonoid of the multiplicative monoid of Br(M) generated by δ, δ−1 and{ri , ei}i∈M is denoted by BrM(M). It is the monoid of monomials in Br(M) and will becalled the Brauer monoid of type M .

Lemma 1.6.2. If i, j ∈ M and ◦i> ◦

j, the following equations hold.

r jeie j = eie j (1.6.25)

eie jei = ei r jei (1.6.26)

e j ri r jei = e jei (1.6.27)

ri r jei r j = r jei r j ri (1.6.28)

ei r jei r j = eie jei (1.6.29)


Proof. By Definition 1.6.1,

r jeie j(1.6.15)= δ−1r jeie jeie j

(1.6.12)= δ−1eie jeie j

(1.6.15)= eie j ,

proving (1.6.25). Therefore

eie jei(1.6.25)= r jeie jei

(1.6.12)= r j r jei r jei

(1.6.1)= ei r jei

giving (1.6.26).It is easy to check that (1.6.27) follows from (1.6.16) and (1.6.2). Also, the identity

(1.6.28) holds as

ri r jei r j(1.6.1)= r j(r j ri r jei)r j

(1.6.13)= r jei r j ri r j r j

(1.6.1)= r jei r j ri .

Finally,

(ei r jei)r j(1.6.26)= ei(e jei r j)

(1.6.17)= eie jei ,

proving (1.6.29).

Similar to the case of type Am (see Remark 1.5.6) there is a natural anti-involution onBr(Cn).

Proposition 1.6.3. The identity map on {δ, ri , ei | i = 0, . . . , n− 1} extends to a uniqueanti-involution on the Brauer algebra Br(M , R,δ).

Proof. It suffices to check the defining relations given in Definition 1.3.1 still hold underthe anti-involution. An easy inspection shows that all relations involved in the definitionare invariant under op, except for (1.6.8), (1.6.11), (1.6.12), (1.6.16), (1.6.17), (1.6.18),(1.6.19), (1.6.21), and (1.6.23). The relation obtained by applying op to (1.6.8) holdsas can be seen by using (1.6.9) followed by (1.6.1) together with (1.6.8). The equality(1.6.16) is the op-dual of (1.6.11). Finally, (1.6.25), (1.6.29), (1.6.18), and (1.6.21)state the op-dual of (1.6.17), (1.6.12), (1.6.19), and (1.6.23), respectively. Hence ourclaim holds.

This anti-involution is denoted by the superscript op, so the map is given by x 7→ xop.Let M ∈ BCFG, Q ∈ ADE be the corresponding type in the first column and the thirdcolumn in Table 1 at the end of chapter 0, σ be the nontrivial diagram automorphism toobtain M in [39], which is

(i) σ =∏n−1

i=1 (i, 2n− i) for Cn in A2n−1,

(ii) σ = (1, 2) for Bn in Dn+1,

(iii) σ = (1, 6)(3,5) for F4 in E6,

(iv) σ = (1, 2,4) for G2 in D4,

andAσ be the subset of σ-invariant admissible root sets ofA under σ.It is well known that the image of {αi | i ∈ Q} under the map αi 7→

∑|σ|t=1σ

t(αi)/|σ|,namely

(

|σ|∑

t=1

σt(αi)/|σ| | i ∈Q

)


consists of the simple roots of type M , where |σ| is the order of σ. We extend the mapto a linear map p : R|Q|→ R|Q|σ = R

|M |, x 7→∑|σ|

t=1σt(x)/|σ|. Let Φ+ be the positive roots

of type Q with respect to {αi}i∈Q and Ψ+ be the positive roots of type M with respect to

{∑|σ|

t=1σt(αi)/ | σ| | i ∈Q}.

We next consider particular sets of mutually orthogonal positive roots in Ψ+, and relatethem to symmetric admissible sets inA .

Definition 1.6.4. Denote byB ′ the collection of all sets of mutually orthogonal roots inΨ+ and by Aσ the subset of σ-invariant elements of A . As p sends positive roots of Φto positive roots of Ψ, it induces a map p :Aσ →B ′ given by p(B) = {p(α) | α ∈ B} forB ∈Aσ. An element ofB ′ will be called admissible if it lies in the image of an admissibleset under p. The set of all admissible elements ofB ′ will be denotedB .

Definition 1.6.5. Suppose that X ⊂Ψ+ is a mutually orthogonal root set. If X is a subsetof some admissible root set, then the minimal admissible set containing X is called theadmissible closure of X , denoted by X or X cl.

Some examples can be seen in Remark 2.4.3 and Remark 3.3.2.

Lemma 1.6.6. Let i ∈ M ∈ BCFG be a node adjacent to another with double or triple bond.Let β ∈Ψ+ be of the maximal height on the W (M)-orbit of βi and r ∈W (M) be the uniqueelement of minimal length such that β = rβi . Let eβ = rei r

−1. Then for each element x inthe stabilizer of β in W (M) (restricted to Ψ+), we have

eβ = xeβ x−1.

Proof. Consider it first for type Cn.There are two W (Cn)-orbits with representatives β0 and β1, and we suppose that β−2 ∈W (Cn)β0 and β−1 ∈W (Cn)β1.Let i = 1 and β = β−1. It is known that the extended Dynkin diagram of type Bn (justconsider the group structure) arises as follows if we add β−1 to the Dynkin diagram oftype Bn.

◦n−1

−1◦

◦n−2· · · · · · ◦

2◦1< ◦

0

According to [5], the stabilizer of β−1 in W (Cn) is generated by reflections in W (Cn)whose roots correspond to nodes in the extended Dynkin diagram of type Bn which arenonadjacent to −1, as we just consider the action on positive roots, the stabilizer of {β1}is

N−1 =

r0, r1, . . . , rn−3, rn−1, r−1�

,

where r−1 = rβ−1= r r1r−1 and r = rn−2rn−1rn−3rn−2 · · · r2r3r1r2r0. Hence it suffices to

prove the lemma holds for each generator of N−1. For r−1,

r−1e−1 = r r1r−1re1r−1 = r r1e1r−1 (1.6.2)= e−1,

e−1r−1 = re1r−1r r1r−1 = re1r1r−1 (1.6.2)= e−1.


For rn−1, we have

rn−1e−1rn−1 = (rn−1rn−2rn−1)rn−3rn−2 · · · e1 · · · rn−2rn−3(rn−1rn−2rn−1)(1.6.7)= rn−2rn−1(rn−2rn−3rn−2 · · · e1 · · · rn−2rn−3rn−2)rn−1rn−2,

by induction, therefore it is reduced to prove that

r1r0e1r0r1 = r0e1r0,

which holds for

(r1r0e1)r0r1(1.6.11)= r0(e1r0r1)

(1.6.16)= r0e1r0.

For r0, we have

r0e−1r0 = r0rn−2rn−1rn−3rn−2 · · · e1 · · · rn−2rn−3rn−1rn−2r0

(1.6.4)= rn−2rn−1 · · · r0r1(r2r0)e1(r0r2)r1r0 · · · rn−1rn−2

(1.6.4)= rn−2rn−1 · · · r0r1r0(r2e1r2)r0r1r0 · · · rn−1rn−2

(1.6.9)= rn−2rn−1 · · · (r0r1r0r1)e2(r1r0r1r0) · · · rn−1rn−2

(1.6.10)= rn−2rn−1 · · · r1r0r1(r0e2r0)r1r0r1 · · · rn−1rn−2

(1.6.5)+(1.6.1)= rn−2rn−1 · · · r1r0(r1e2r1)r0r1 · · · rn−1rn−2

(1.6.9)= rn−2rn−1 · · · r1(r0r2)e1(r2r0)r1 · · · rn−1rn−2

(1.6.4)= rn−2rn−1 · · · r1r2r0e1r0r2r1 · · · rn−1rn−2 = e−1.

For rn−3, we have

rn−3e−1rn−3 = (rn−3rn−2rn−1rn−3)rn−2 · · · e1 · · · rn−2(rn−3rn−1rn−2rn−3)(1.6.4)+(1.6.7)= rn−2rn−3(rn−2rn−1rn−2) · · · e1 · · · (rn−2rn−1rn−2)rn−3rn−2

(1.6.7)= rn−2rn−3rn−1rn−2(rn−1 · · · e1 · · · rn−1)rn−2rn−1rn−3rn−2

(1.6.4)+(1.6.5)= rn−2rn−3rn−1rn−2 · · · e1 · · · (rn−1rn−1)rn−2rn−1rn−3rn−2

(1.6.1)= rn−2(rn−3rn−1)rn−2 · · · e1 · · · rn−2(rn−1rn−3)rn−2

(1.6.1)= rn−2rn−1rn−3rn−2 · · · e1 · · · rn−2rn−3rn−1rn−2 = e−1.

For any ri with 1≤ i ≤ n− 4, we can prove the required equality by induction as rn−3.Let i = 0 and β = β−2. It is known that the extended Dynkin diagram of type Cn arises ifwe add β−2 to the Dynkin diagram of type Cn.

◦−2

> ◦n−1

◦n−2· · · · · · ◦

2◦1< ◦

0

Similarly, the stabilizer of {β−2} is

N−2 =

r0, r1, . . . , rn−3, rn−2, r−2�

,


where r−2 = rβ−2= r r1r−1 and r = rn−1rn−2 · · · r3r2r1.

Hence it also suffices to prove the lemma holds for each generator of N−2.It can be easily verified that r−2e−2 = e−2r−2 = e−2 for r−2.For rn−2, we have

r0e−2r0 = (r0rn−1rn−2 · · · r2)r1e0r1(r2 · · · rn−2rn−1r0)(1.6.4)= rn−1rn−2 · · · r2r0r1e0r1r0r2 · · · rn−2rn−1

(1.6.1)= rn−1rn−2 · · · r2r1(r1r0r1e0)r1r0r1r1r2 · · · rn−2rn−1

(1.6.13)= rn−1rn−2 · · · r2r1e0(r1r0r1r1r0r1)r1r2 · · · rn−2rn−1

(1.6.1)= rn−1rn−2 · · · r2r1e0r1r2 · · · rn−2rn−1 = e−2.

For rn−2, we have

rn−2e−2rn−2 = (rn−2rn−1rn−2) · · · r2r1e0r1r2 · · · (rn−2rn−1rn−2)(1.6.7)= rn−1rn−2(rn−1 · · · r2r1e0r1r2 · · · )rn−1rn−2rn−1

(1.6.4)+(1.6.5)= rn−1rn−2 · · · r3r2r1e0r1r2 · · · (rn−1rn−1)rn−2rn−1

(1.6.4)+(1.6.5)= rn−1rn−2 · · · r3r2r1e0r1r2 · · · rn−2rn−1 = e−2.

Therefore we prove the lemma for M = Cn.If M = Bn, the argument is nearly the same as that for Cn, even the formulas for e−1 ande−2 are kept. The only difference is choosing formulas from (1.6.10)–(1.6.17) carefullyfor the alternative relations between 0 and 1.For M = F4 and G2, it is quite similar with the corresponding extended Dynkin diagramschanged as below.

◦−1

◦1

◦2< ◦

3◦4

◦1

◦2< ◦

3◦4

◦−2

◦−1

◦0< ◦

1

◦0< ◦

1◦−2

In general, we have

weβw−1 = ewβ , (1.6.30)

for w ∈ W (M) and β a root of W (M) for M ∈ BCFG, which is well defined due to theLemma 1.6.6. Note that eβ = e−β .

We continue by discussing some desirable properties of the Brauer algebra Br(M) forM ∈ BCFG. First of all, for any disjoint union J of diagrams of simply laced type Q, theBrauer algebra is defined as the direct product of the Brauer algebras whose types are thecomponents of J . The next result states that parabolic subalgebras of Br(M) behave wellwith the hypothesis that the subalgebra relation in Table 1 is obtained, and it answersQuestion 1 at the end of Chapter 0.

1.7 BMW ALGEBRAS OF SIMPLY LACED TYPE 23

Proposition 1.6.7. Let J be a set of nodes of the Dynkin diagram M ∈ BCFG. Then theparabolic subalgebra of the Brauer algebra Br(M), that is, the subalgebra generated by{r j , e j} j∈J , is isomorphic to the Brauer algebra of type J.

Proof. We just do the example for M = Cn, the remaining cases can be verified by similararguments.In view of induction on n− |J | and restriction to connected components of J , it sufficesto prove the result for J = {1, . . . , n − 1} and for J = {0, . . . , n − 2}. In the formercase, the type is An−1 and the statement follows from the observation that the symmetricdiagrams without strands crossing the vertical line through the middle of the segmentsconnecting the dots (n, 1) and (n + 1,1) are equal in number to the Brauer diagramson the 2n nodes (realized to the left of the vertical line). In the latter case, the type isCn−1 and the statement follows from the observation that the symmetric diagrams withvertical strands from (1, 1) to (1,0) and from (2n, 1) to (2n, 0) are equal in number tothe symmetric diagrams related to BrM(Cn−1).

In [33], the nontrivial diagram automorphisms on Dynkin diagrams were generalizedas admissible partitions. We introduce the definition and a theorem from [33]. In thisthesis, we apply the same idea to obtain the Brauer algebras of type In

2, H3, H4 fromBrauer algebras of type An−1, D6, E8, respectively.

Definition 1.6.8. Let (W, S) be a Coxeter system of type M . If J is a spherical subset ofS, we write wJ for the longest element of WJ , the parabolic subgroup of W generated byJ . If P is a partition of S all of whose parts are spherical , we denote CW (P) the subgroupof W generated by all wJ for J ∈ P.A partition P of S is called an admissible partition if, for each part A ∈ P, the Coxeterdiagram restricted to A is spherical and, for each element w ∈ CW (P), either l(rw)< l(w)for all r ∈ A or l(rw)> l(w) for all r ∈ A.

Theorem 1.6.9. Let P be an admissible partition of S and write SP = {wA | A ∈ P}. Thenthe pair (CW (P), SP) is a Coxeter system. Its type is the Coxeter diagram MP on SP whose A,B-entry is the order of wAwB in W.

Remark 1.6.10. If we write H2 = I52 and have proved the conclusion in Table 1, the

commutative diagram below follows.

Br(H2) //

��

Br(H3)

��

// Br(H4)

��Br(A4) // Br(D6) // Br(E8)

Since the homomorphisms in the second rows and the vertical homomorphisms are in-jective, it follows that the homomorphisms in the first rows are also injective. Thereforethe parabolic proposition for Br(H3) and Br(H4) can be verified.

1.7 BMW algebras of simply laced type

Definition 1.7.1. Let Q be a simply laced Coxeter diagram of rank n. The BMW algebraof type Q is the algebra, denoted by B(Q), with ground field Q(l,δ), where l and δ are


transcendental and algebraically independent over Q, whose presentation is given ongenerators gi and ei (i = 1,2,. . ., n) by the following relations

gi g j = g j gi for i⊥ j (1.7.1)

gi g j gi = g j gi g j for i ∼ j (1.7.2)

mei = l(g2i +mgi − 1) for any i (1.7.3)

giei = l−1ei for any i (1.7.4)

ei g jei = lei for i ∼ j (1.7.5)

where m = (l − l−1)/(1− δ), and i ∼ j means that i and j are adjacent in the Dynkindiagram Q, and i⊥ j indicates that they are distinct and non-adjacent.

Remark 1.7.2. It is known there is a natural homomorphism of rings from the BMWalgebra of type Q to the Brauer algebra of the same type induced on the generators bygi 7→ Ri and ei 7→ Ei with l = 1. In [10], [12], [17], it is proved that the rewritten formin Proposition 1.3.3 gives a basis of the BMW algebra by changing Ri to gi and Ei to ei .After this thesis, one want to know whether we can obtain the BMW algebras of non-simply laced types from simply laced types. In fact, we have tried this but failed becauseit is difficult to control the BMW algebras since the generators of BMW algebras cannotgive a good monoid as Brauer algebras. For example, by the Definition 1.7.1 for Br(A3),it follows that

(g1 g3)2 = m2 g1 g3 + l−2m2e1e3 + 1−m(g1 + g3) + l−1m(e1 + e3) + l−1m2(e1 g3 + g1e3),

where some new approach should be created to deal with the last three terms to findtype C2 from type A3.

2TYPE Cn

This chapter is based on our paper [14].

2.1 Introduction

It is well known that the Coxeter group of type Cn arises from the Coxeter group of typeA2n−1 as the subgroup of all elements fixed by a Coxeter diagram automorphism. Crisp[19] showed that the Artin group of type Cn arises in a similar fashion from the Artingroup of type A2n−1. In this Chapter, we study the subalgebra of the Brauer algebraBr(A2n−1) of type A2n−1 (that is, the classical Brauer algebra on 2n strands) spanned byBrauer diagrams that are fixed by the symmetry corresponding to this diagram automor-phism (see Definition 1.3.1). Such diagrams will be called symmetric. First, in Definition1.6.1, we define the Brauer algebra of type Cn, denoted by Br(Cn), in terms of generatorsand relations depending solely on the Dynkin diagram below.

Cn = ◦n−1

◦n−2· · · · · · ◦

2◦1< ◦

0

The distinguished generators of Br(Cn) are the involutions r0, . . . , rn−1 and the quasi-idempotents e0, . . . , en−1 (here, a quasi-idempotent is an element that is an idempotentup to a scalar multiple). Each defining relation concerns at most two indices, say i andj, and is determined by the diagram induced by Cn on {i, j}. The group algebra ofthe Coxeter group of type Cn is obtained by taking the quotient of the Brauer algebraof type Cn by the ideal generated by all quasi-idempotents ei . It is isomorphic to thesubalgebra generated by all ri . The subalgebra generated by all ei (i = 0, . . . , n− 1) isisomorphic to the Temperley–Lieb algebra of type Bn defined by tom Dieck in [20], [21].The main result states that the algebra Br(Cn) is isomorphic to the subalgebra SBr(A2n−1)of the Brauer algebra Br(A2n−1) linearly spanned by symmetric diagrams. In order todistinguish them from those of Br(Cn), the canonical generators of the Brauer algebraof type A2n−1 are denoted by R1, . . . , R2n−1, E1, . . . , E2n−1 instead of the usual lower caseletters (see Definition 1.3.1). Although our formal set-up is slightly more general, thealgebras considered are mostly defined over the integral group ring Z[δ±1].

Theorem 2.1.1. There exists a Z[δ±1]-algebra isomorphism

φ : Br(Cn)−→ SBr(A2n−1)

26 TYPE Cn

determined by φ(r0) = Rn, φ(ri) = Rn−iRn+i , φ(e0) = En, and φ(ei) = En−i En+i , for0 < i < n. In particular, the algebra Br(Cn) is free over Z[δ±1] of rank a2n, where an isdefined by a0 = a1 = 1 and, for n> 1, the recursion

an = an−1 + 2(n− 1)an−2.

A closed formula for the rank of Br(Cn) is

a2n =n∑

i=0

∑

p+2q=i

n!

p!q!(n− i)!

2

2n−i (n− i)!. (2.1.1)

A table of an for some small n is provided below.

n 0 1 2 3 4 5 6 7 8an 1 1 3 7 25 81 331 1303 5937

The chapter is structured as follows. In Section 2.2, we proveφ appearing in Theorem2.1.1 is an algebra homomorphism and the recursion formula of an. In Section 2.3, wederive elementary properties of Br(Cn). Next, in Section 2.4, we prove that the imageof φ is precisely the symmetric diagram subalgebra SBr(A2n−1) of Br(A2n−1). In Section2.5 we study symmetric diagrams, the related algebra SBr(A2n−1), and the action of themonoid of all monomials of Br(Cn) on certain orthogonal root sets and a normal form formonomials in Br(Cn). With these results, we are able to prove Theorem 2.1.1. Finally, inSection 2.6, we establish cellularity (in the sense of [26]) of the newly introduced Braueralgebras and derive some further properties.

We finish this introduction by illustrating our results with the first interesting case:n= 2. Consider the classical Brauer algebra Br(A3). The corresponding Brauer diagramsconsist of four nodes at the top and four at the bottom together with a complete matchingbetween these eight nodes. See Figure 1.1 for interpretations of Ri and Ei (i = 1,2, 3).A Brauer diagram is called symmetric if the complete matching is not altered by thereflection of the plane whose mirror is the vertical central axis of the diagram. Clearly,e1 := E1E3, e0 := E2, r1 := R1R3, and r0 := R2 represent symmetric diagrams. Ourmain theorem implies that the subalgebra of Br(A3) generated by these diagrams has apresentation on these four generators by the relations given in Definition 1.6.1 for n= 2,and moreover it coincides with SBr(A3), the linear span of all symmetric diagrams. Infact, it is free and spanned by the following 25 monomials.

1, r0, r1, r0r1, r1r0, r1r0r1, r0r1r0r1, r0r1r0,

{1, r1}e0{1, r1r0r1}{1, r1},{1, r0, e0}e1{1, r0, e0}.

Here, AxBC for subsets A, B, C and an element x of SBr(A3), indicates the sequence ofall elements ax bc with a ∈ A, b ∈ B, c ∈ C , and similarly for AxB. The first eight, givenon the top line, span a subalgebra isomorphic to the group algebra of the Weyl groupof type C2. This is in accordance with the construction of SBr(A3) and the fact that theWeyl group of type C2 occurs in the Weyl group of type A3 as the subgroup of elements

2.2 THE HOMOMORPHISM φ AND THE NUMBER an 27

fixed by a Coxeter diagram automorphism. The two-sided ideal of SBr(A3) generatedby e1 is spanned by the 9 monomials on the bottom line. Also, the complement in theideal generated by e0 and e1 of the ideal generated by e1 is spanned by the 8 monomialson the middle line. This division of the 25 spanning monomials into three parts alongthe above lines is strongly related to the cellular structure of SBr(A3). The subalgebraof SBr(A3) generated by e0 and e1 has dimension 6 and is isomorphic to the Temperley–Lieb algebra of type B2 introduced by tom Dieck [21]. For these (and other) reasons, wename SBr(A3) the Brauer algebra of type C2. Remarkably, the Temperley–Lieb algebra oftype B2 defined by Graham [25] is 7-dimensional and tom Dieck’s version is a quotientalgebra thereof, but we have not found a natural extension of Graham’s algebra to anobject deserving the name Brauer algebra of type C2.

2.2 The homomorphism φ and the number an

The definition of Br(Cn) is given in Definition 1.6.1. In this section, we prove that themap φ appearing in Theorem 2.1.1 is an algebra homomorphism and we establish therecursion formula of an in 2.1.1. All rings and algebras given are unital and associative.

Observe that, for a distinguished invertible element δ, the ring R can be viewed asa Z[δ±1]-algebra and that Br(Cn, R,δ) ∼= Br(Cn) ⊗Z[δ±1] R. As a direct consequence ofthe above definition, the submonoid of BrM(Cn) generated by {ri | i = 0, . . . , n− 1} isisomorphic to the Weyl group W (Cn) of type Cn.

Let us recall from Definition 1.3.1 the definition of a Brauer algebra of simply lacedCoxeter type Q. In order to avoid confusion with the above generators, the symbols inDefinition 1.3.1 have been capitalized. For any Q, the algebra Br(Q) is free over Z[δ±1].The classical Brauer algebra on m+ 1 strands is obtained when Q = Am.

The map σ on the graph (or Coxeter type) Am given by σ(i) = m+ 1− i is the singlenontrivial automorphism of this graph. As the presentation of Br(Q) merely depends onthe graph Q, the map σ induces an automorphism of Br(Am), which will also be denotedby σ. This involutory automorphism is determined by its behaviour on the generators:

σ(Ri) = Rm+1−i , σ(Ei) = Em+1−i .

The automorphism σ may be viewed simply as a reflection of the corresponding dia-gram about its central vertical axis.

Definition 2.2.1. Suppose D1 and D2 are diagrams in Br(Am). The diagram D1 is sym-metric to the diagram D2 if D2 is the diagram obtained by taking the reflection of D1about its central vertical axis. If D1 = D2, then we say D1 is a symmetric diagram.

Hence a diagram (that is, a monomial in Ri and Ei) of Br(Am) is σ- invariant if and onlyif it is symmetric about its central vertical axis. A monomial in BrM(Am) is fixed by σ ifand only if it represents a symmetric diagram.

Lemma 2.2.2. Let m = 2n− 1, for some n ∈ N. The number of symmetric diagrams (withrespect to σ) in BrM(Am) is equal to a2n, where an satisfies a0 = a1 = 1 and the recursion

an = an−1 + 2(n− 1)an−2.

28 TYPE Cn

Proof. Fix two sets X and Y , say, of size n and a permutation τ of X ∪ Y of order 2interchanging X and Y . We define an as the number of perfect matchings on X ∪ Y thatare τ-invariant (that is, if {a, b} ⊆ X ∪ Y belongs to the matching, so does {τ(a),τ(b)}).Identifying X with the set of dots left of the vertical axis of symmetry, Y with the set ofdots to the right, and τ with the permutation induced by σ, we see that that a2n is thenumber of symmetric diagrams in BrM(Am).

It is obvious that a0 = a1 = 1. Fix a ∈ X . The number of perfect τ-invariant matchingscontaining {a,τ(a)} is equal to an−1.

Suppose that we have a perfect τ-invariant matching of X ∪ Y containing {a, b} withb 6= τ(a). Then {τ(a),τ(b)} is a second pair belonging to the matching. The matchinginduces an−2 number of perfect τ-invariant matchings on (X ∪ Y ) \ {a, b,τ(a),τ(b)}. Asthere are 2n− 2 choices of b, we find an = an−1 + (2n− 2)an−2.

Corollary 2.2.3. The linear span SBr(A2n−1) of symmetric diagrams is a Z[δ±1]-subalgebraof Br(A2n−1). It is free over Z[δ±1] of rank a2n.

Observe that Rn, RiR2n−i , En, and Ei E2n−i are fixed under σ for all i ∈ {1, . . . , n}. Thusthe image of the map φ of Lemma 2.2.4 below lies in SBr(A2n−1).

Lemma 2.2.4. The following map determines a Z[δ±1]–algebra homomorphismφ : Br(Cn)→SBr(A2n−1).

φ(r0) = Rn, φ(ri) = Rn−iRn+i ,φ(e0) = En and φ(ei) = En−i En+i , for 0< i < n.

Proof. It suffices to verify thatφ preserves the defining relations given in Definition 1.6.1.We demonstrate this for some of the relations (1.6.1)–(1.6.17), and leave the rest as anexercise for the reader.For (1.6.12):

φ(r1)φ(e0)φ(r1)φ(e0) = Rn−1(Rn+1EnRn+1)Rn−1En

(1.3.9)= Rn−1RnEn+1(RnRn−1En)(1.3.8)= Rn−1RnEn+1En−1En

(1.3.6)+(1.3.8)= EnEn−1En+1En

= φ(e0)φ(e1)φ(e0).

For (1.6.13):

φ(r1)φ(r0)φ(r1)φ(e0) = Rn+1(Rn−1RnRn−1)Rn+1En

(1.3.7)= Rn+1RnRn−1(RnRn+1En)

(1.3.8)+(1.5.1)= Rn+1RnRn−1En+1RnRn+1

(1.3.5)= Rn+1RnEn+1Rn−1RnRn+1

(1.3.8)+(1.5.1)= EnRn+1RnRn−1RnRn+1

(1.3.7)= EnRn+1Rn−1RnRn−1Rn+1

= φ(e0)φ(r1)φ(r0)φ(r1).

2.3 ELEMENTARY PROPERTIES OF TYPE C ALGEBRAS 29

For (1.6.17):

φ(e1)φ(e0)φ(r1)(1.3.4)= En−1(En+1EnRn+1)Rn−1

(1.5.4)= En−1En+1RnRn−1

(1.3.6)+(1.5.1)= En+1En−1En = φ(e1)φ(e0).

At this point, we have explained the algebras and the map φ occurring in Theorem2.1.1. The surjectivity of φ will be proved in Proposition 2.4.13 and its injectivity at theend of Section 2.5.

2.3 Elementary properties of type C algebras

For each i ∈ {1, . . . , n}, we define the following two elements of BrM(Cn).

yi := ri−1ri−2 · · · r1r0r1 · · · ri−2ri−1, (2.3.1)

zi := ri−1ri−2 · · · r1e0r1 · · · ri−2ri−1. (2.3.2)

Proposition 2.3.1. Let n≥ 2 and i ∈ {2, . . . , n} and consider elements in BrM(Cn).

(i) ei , ri , yi , and zi commute with each of r j and e j for 0≤ j ≤ i− 2.

(ii) yi and zi commute with y j and z j for each j ∈ {1, . . . , n}.

Proof. (i). By Definition 1.6.1, both ei and ri commute with each element of{r0, . . . , ri−2, e0, . . . , ei−2}.

In order to prove that yi commutes with the indicated elements, we first establish theclaim that yi+2 commutes with ri and ei , for 0≤ i ≤ n− 2.

If i = 0, the claim follows from (1.6.10) and (1.6.13), respectively. If i > 0, we have

yi+2ri = ri+1ri · · · r1r0r1 · · · ri ri+1ri(1.6.7)= ri+1ri · · · r1r0r1 · · · ri−1ri+1ri ri+1

(1.6.4)= ri+1ri ri+1ri−1 · · · r1r0r1 · · · ri ri+1

(1.6.7)= ri ri+1ri · · · r1r0r1 · · · ri−1ri ri+1 = ri yi+2,

and

yi+2ei = ri+1ri · · · r1r0r1 · · · ri ri+1ei(1.6.9)= ri+1ri · · · r1r0r1 · · · ri−1ei+1ri ri+1

(1.6.5)= ri+1riei+1ri−1 · · · r1r0r1 · · · ri ri+1

(1.6.9)= ei ri+1ri · · · r1r0r1 · · · ri−1ri ri+1 = ei yi+2.

Now, for arbitrary i and all 0 ≤ j ≤ i − 2, using yi = ri−1 · · · r j+2 y j+2r j+2 · · · ri−1 and(1.6.4), we find

yi r j = ri−1 · · · r j+2 y j+2r j+2 · · · ri−1r j = ri−1 · · · r j+2 y j+2r j r j+2 · · · ri−1

= ri−1 · · · r j+2r j y j+2r j+2 · · · ri−1 = r j ri−1 · · · r j+2 y j+2r j+2 · · · ri−1

= r j yi

30 TYPE Cn

Similarly for e j instead of r j , by use of (1.6.5).An analogous argument can be used for zi . Again, it suffices to show that zi+2 com-

mutes with ri and ei . For i = 0 we verify

z2e0 = (r1e0r1)e0(1.6.12)+(1.6.29)

= e0(r1e0r1) = e0z2,

z2r0 = (r1e0r1)r0(1.6.1)= r1e0r1r0r1r1

(1.6.13)+(1.6.1)= r0(r1e0r1) = e0z2.

For i > 0, it is straightforward to show that zi+2ri = rizi+2, using (1.6.4) and (1.6.7), andzi+2ei = eizi+2, by using a rearrangement of (1.6.9). Thus, an argument similar to theabove proves that zi r j = r jzi and zie j = e jzi , for any i and all 0≤ j ≤ i− 2.

As yi and zi are conjugates of r0 and e0 by the same Coxeter group element, it followsfrom (1.6.2) that they commute. It remains to verify that yi and zi commute with yi−1and zi−1. But the latter two elements are products of generators from {r0, . . . , ri−2, e0, . . . , ei−2},which are known to commute with yi and zi by (i) and (ii). This finishes the proof of (i)and (ii).

The following lemma can be verified easily by counting the number of diagrams.

Lemma 2.3.2. The Z-subalgebra generated by {ri , ei}n−1i=1 and δ2 in Br(Cn) is isomorphic to

Br(An−1).

The following ensures that Br(Cn) is of finite rank over Z[δ±1].

Proposition 2.3.3. Let Xn = {1, rn−1, en−1, yn−1, zn−1}. For the monoid BrM(Bn), we have

BrM(Bn) = BrM(Bn−1)XnBrM(Bn−1),

therefore Br(Bn) is of finite rank over Z[δ±1].

Proof. By induction, we suppose BrM(Bn−1) = BrM(Bn−2)Xn−1BrM(Bn−2). To prove thisproposition, it is sufficient to prove that XnBrM(Bn−1)Xn ⊂ BrM(Bn−1)XnBrM(Bn−1), whichis equivalent to prove that XnXn−1Xn ⊂ BrM(Bn−1) for Lemma 2.3.1.

Considering Proposition 1.6.3 and Lemma 2.3.1, therefore we can see that to provethe above is equivalent to prove that the followings are in BrM(Bn−1)XnBrM(Bn−1) whichare

r2n−1, rn−1en−1, rn−1 yn, rn−1zn,

rn−1rn−2rn−1, rn−1rn−2en−1, rn−1en−2rn−1, rn−1en−2en−1,e2

n−1, en−1rn−2en−1, en−1en−2en−1, y2n , yn−1zn, z2

n ,rn−1 yn−1rn−1, rn−1zn−1en−1, en−1 yn,en−1zn, en−1 yn−1en−1, en−1zn−1en−1.By equations (1.6.1)-(1.6.17) in Definition 1.6.1 and Lemma 2.3.2, we can see that

the above elements except the last 6 elements are in BrM(Bn−1)XnBrM(Bn−1).It is obvious that yn = rn−1 yn−1rn−1 and zn = rn−1zn−1rn−1 are in BrM(Bn−1)XnBrM(Bn−1).

2.4 SURJECTIVITY OF φ 31

By induction we suppose that ei−1 yi = ei−1 yi−1 for i < n. Then we have that

en−1 yn−1 = en−1rn−1rn−2 yn−2rn−2rn−1

(1.6.2)= en−1rn−2 yn−2rn−2rn−1

(1.6.1)= en−1rn−2r2

n−1 yn−2rn−2rn−1

(1.5.1)+2.3.1= en−1en−2rn−1 yn−2rn−2rn−1

2.3.1= en−1en−2 yn−2rn−1rn−2rn−1

(1.6.7)= en−1en−2r2

n−2 yn−2rn−2rn−1rn−2

induct ion= en−1en−2 yn−1rn−1rn−2

2.3.1= en−1en−2rn−1 yn−2rn−2

(1.5.4)= en−1rn−2 yn−2rn−2 = en−1 yn−1.

The similar argument can be applied to prove en−1zn = en−1zn−1.By induction we suppose that ei−1 yi−1ei−1 = δei−1 for i < n. Then we have that

en−1 yn−1en−1 = en−1rn−2 yn−2rn−2en−1

(1.6.1)= en−1rn−2r2

n−1 yn−2rn−2en−1

2.3.1= en−1rn−2rn−1 yn−2rn−1rn−2en−1

(1.6.10)+2.3.1= en−1en−2 yn−2en−2en−1

induct ion= δen−1en−2en−1

(1.6.9)= δen−1.

The similar argument can be used for en−1znen−1 = δen−1.Therefore our proposition holds.

2.4 Surjectivity of φ

The goal of this section is to exhibit a collection of admissible sets on which BrM(Cn) actsas well as to prove that the map φ : Br(Cn) −→ SBr(A2n−1) introduced in Theorem 2.1.1is surjective. To this end, we first construct the root system of type Cn in terms of σ-fixedvectors in the reflection space for W (A2n−1) spanned by the root system Φ of Definition1.5.3. Notice that the restriction of φ to the submonoid W (Cn) of BrM(Cn) generated bythe ri (isomorphic, as the notation suggests, to the Coxeter group of type Cn) is knownto be injective (see, for instance [33]), with image the centralizer of σ in the submonoidW (A2n−1) of BrM(A2n−1).

We adopt the notation of Section 1.5 with m = 2n− 1, and will use the root systemΦ, the collection of admissible setsA , and the admissible set action of BrM(A2n−1) onAof Definition 1.3.2. We let the involution σ act on the set Φ+ of positive roots of Φ in thefollowing way, where αi are as in Definition 1.5.3. For Σciαi ∈ Φ+ we decree

σ(Σciαi) = Σciα2n−i (1≤ i < 2n).

32 TYPE Cn

This map σ induces the permutation of the simple roots corresponding to the nontriv-ial automorphism of the Coxeter diagram A2n−1. This permutation can be extended tothe linear transformation of R2n, again denoted σ, determined by σ(εi) = −ε2n+1−i .(Indeed, this transformation satisfies σ(αi) = α2n−i for each i ∈ {1, . . . , 2n− 1}). Thevectors fixed by σ form an n-dimensional subspace, to be denoted R2n

σ , of the (2n− 1)-dimensional subspace of R2n spanned by Φ.

Definition 2.4.1. Let p : R2n → R2nσ be the orthogonal projection from R2n onto R2n

σ ,that is, p(x) = (x +σ(x))/2 for x ∈ R2n. Let α ∈ Φ. Then p(α) = α is of squared norm2 if σ(α) = α and p(α) = 1

2(α+ σ(α)) is of squared norm 1 if σ(α) 6= α. The image

Ψ = p(Φ) of Φ under p is a root system of type Cn with simple roots β0 = p(αn) = αnand βi = p(αn−i) = p(αn+i) for i = 1, . . . , n − 1. It is contained in R2n

σ and spans it.Of course, Ψ+ will be understood to be the half of Ψ lying in the cone spanned by βi(i = 0, . . . , n−1). Given α ∈ Φ we write Rα for the orthogonal reflection on R2n with rootα. Given β ∈ Ψ we write rβ for the orthogonal reflection on R2n

σ with root β . We mayidentify Ri and r j with Rαi

and rβ j, respectively.

Recall that φ(r0) = Rn and φ(r j) = Rn− jRn+ j for j ∈ {1, . . . , n− 1}. The followinglemma collects some elementary properties of the maps introduced so far, the proofs ofwhich are omitted.

Lemma 2.4.2. The maps σ, p, and the restriction of φ to W (Cn) satisfy the followingproperties for each w ∈W (Cn), α ∈ Φ, and β ∈Ψ.

(i) σφ(w) = φ(w)σ.

(ii) φ(w)x = wx if x ∈ R2nσ .

(iii) p(φ(w)α) = wp(α).

(iv) φ(rβ) =∏

γ∈p−1(β) Rγ and φ(eβ) =∏

γ∈p−1(β) Eγ. Here, the set p−1(β) has cardinality1 or 2, according to β ∈W (Cn)β0 or β ∈W (Cn)β1.

Remark 2.4.3. Recall the Definition 1.6.4. Not all sets of mutually orthogonal roots inΨ+ are admissible. For instance Y = {β1,β1+β0} (two mutually orthogonal short roots)belongs to B ′ (for n = 2) but not to B . For, if X ∈ Aσ would be such that p(X ) = Y ,then X should contain α1 and α3 as well as α1 +α2 and α2 +α3; but these roots are notmutually orthogonal. On the other hand, for n ≥ 4, the unordered pair {β1,β3} fromanother W (Cn)-orbit of mutually orthogonal short roots, is the image of the admissibleset {αn−1,αn+1,αn−3,αn+3} and so belongs toB .

Also {β0, 2β1+β0} (two mutually orthogonal long roots) does belong toB (for n= 2)as it coincides with p(X ), where X = {αn,αn−1 +αn +αn+1}.

Proposition 2.4.4. The monoid BrM(Cn) acts on Aσ under the composition of the admis-sible set action and φ.

Proof. It suffices to prove that Aσ is closed under the action of φ(BrM(Cn)). It is easyto see that σ(a)σ(B) = σ(aB), for a ∈ BrM(A2n−1) and B ∈ A . Consequently, if a ∈BrM(A2n−1)σ and B ∈ Aσ, then it follows that aB = σ(a)σ(B) = σ(aB). This showsaB ∈Aσ. As φ(BrM(Cn))⊆ BrM(A2n−1)σ, the proposition follows.


Proposition 2.4.5. The map p :Aσ→B is bijective and W (Cn)-equivariant, so p(φ(w)X ) =wp(X ) for X ∈Aσ and w ∈W (Cn).

Proof. The map p is surjective by definition of B . Let Y ∈ B and X ∈ p−1(Y ). Ifβ ∈ Y , then there is α ∈ X such that β = p(α). As X ∈ Aσ, it follows that σα ∈ X , soX = {α ∈ Φ | p(α) ∈ Y } is uniquely determined by Y . This shows that p is injective.

Finally, if in addition, w ∈ W (Cn), then φ(w)X ∈ Aσ by Proposition 2.4.4, and, byLemma 2.4.2,

p(φ(w)X ) = {p(φ(w)α) | α ∈ X }= w{p(α) | α ∈ X }= wp(X ).

Consider a positive root β and a node i of the Dynkin diagram Cn. Recall (1.6.30).For (1.6.30), we can reinterpret the element yi of (2.3.1) as rγ and zi of (2.3.2) as eγ,

where γ= β0+2β1+ · · ·+2βi−1. Proposition 2.3.1 shows that, for each i ∈ {0, . . . , n−1},

b(i) =i∏

k=1

zk (2.4.1)

is well defined. The admissible set

Bi = p−1 �{β0,β0 + 2β1, . . . ,β0 + 2β1 + · · ·+ 2βi−1}�

(2.4.2)

inAσ is both the top and the bottom of φ(b(i)). In other words, the symmetric diagramof φ(b(i)) has horizontal strands from (n+ 1− j, 1) to (n+ j, 1) and from (n+ 1− j, 0)to (n+ j, 0) for each j ∈ {1, . . . , i}. This is the special case p = i of the top displayed inFigure 2.1. Later, below (2.4.5), we will use this fact.

The Bi (i = 0, . . . , n) are a complete set of W (A2n−1)-orbit representatives in A .Moreover, φ(b(i)) has height 0.

Proposition 2.4.6. Let β and γ be positive roots of Ψ.

(i) eβ rβ = rβ eβ = eβ , e2β = δ

2eβ if β is short, and e2β = δeβ if β is long.

(ii) If (β ,γ) =±1 and β and γ are short, then eβ rγeβ = eβ , rβ rγeβ = eγrβ rγ = eγeβ , andeβ eγeβ = eβ .

(iii) If (β ,γ) = ±1 with β short and γ long, then the equations (1.6.10)–(1.6.17) and(1.6.25)–(1.6.29) still hold with the subscripts 1 and 0 replaced by β and γ, respec-tively.

(iv) If (β ,γ) = 0 and β and γ are both long, then eβ eγ = eβ eγ.

(v) If (β ,γ) = 0 and β and γ are both short, and there exists a long positive root α suchthat β = γ + α or β = γ − α, then eβ eγ = δrαeγ or eγeβ = δrαeβ . In each case,eβ eγ 6= eγeβ .

Proof. The assertions are easily proved after reduction to simple cases using Lemma1.6.6. We illustrate the argument by treating (v) now in greater detail.

Up to an interchange of β and γ, there exists w ∈ W (Cn) such that wβ0 = α and

wβ1 = γ and β = γ+α= rαγ. Now eβ eγ = rαeγrαeγ = wr0e1r0e1w−1 (1.6.14)= δwr0e1w−1 =

δrαeγ. The inequality stated at the end of part (v) follows from the inspection of sym-metric Brauer diagrams in the image of φ.

34 TYPE Cn

As a consequence of Proposition 2.4.6(iv) and (1.6.6), the product of eβ , for β run-ning over the members of an admissible set, does not depend on the order. Therefore,for each B ∈B , we may define

eB =∏

β∈B

eβ , (2.4.3)

This is very similar to the definition of EB in (1.3.11). Part (v) of the propositionshows that it is essential that B be admissible for a set B of orthogonal roots to define aproduct as in (2.4.3): the elements eβ and eγ for β and γmutually orthogonal short rootsdo not commute. This is consistent with Remark 2.4.3, in that the above set {β ,γ} is notadmissible. The admissible sets of size 2 belong to W (Cn)-orbits of pairs of mutuallyorthogonal simple positive roots and so behave well in (2.4.3) thanks to (1.6.6).

Lemma 2.4.7. Let B ∈Aσ. Then EB = φ(ep(B)) and hence EB ∈ φ(Br(Cn)).

Proof. By Lemmas 2.4.2(iv) and 2.2.4,

EB =∏

α∈B

Eα =∏

β∈p(B)

Ep−1(β) =∏

β∈p(B)

φ(eβ) = φ

∏

β∈p(B)

eβ

= φ(ep(B)),

As W (Cn) is a subgroup of the monoid BrM(Cn), it also acts on Aσ (cf. Proposition2.4.4). We will show that the admissible sets defined below are orbit representatives forthis action.

Definition 2.4.8. For i and p with 0≤ p ≤ i ≤ n and i− p even, write

ei,p = ep+1ep+3 · · · ei−1, (2.4.4)

and

Bi,p = ep+1ep+3 · · · ei−1Bi .

In addition, for p′ ∈ N with 0≤ p′ ≤ i < n and i − p′ even, we write

bp,i,p′ = ei,p b(i)ei,p′ , (2.4.5)

where b(i) is as defined in (2.4.1).

Observe that Bi,i = Bi . The admissible set Bi,p is pictured in Figure 2.1 as the top of aBrauer diagram.

Lemma 2.4.9. If 0 < p, p′ < i with i − p and i − p′ even, then φ(bp,i,p′) is a diagram ofheight 0 with top Bi,p and bottom Bi,p′ . Moreover, φ(bp,i,p′) = aBi,p ,Bi,p′

.

Proof. This is an easy verification involving the left and right monoid actions of Proposi-tion 2.4.4 (use that eop

i,p′ = ei,p′).


n n+1 n+p n+i-1 n+i 2n1 n-i+1 n-i+2 n-p+1

Figure 2.1: The admissible set Bi,p.

Definition 2.4.10. Let i ∈ {0, . . . , n}. Define E(i) to be the idempotent in BrM(A2n−1)corresponding to b(i), that is, E(i) := δ−iφ(b(i)). Then E(i) = φ(b(i)), where b(i) :=δ−i b(i). Furthermore, we write Ki instead of KBi

as introduced in Definition 1.5.9.

Observe that Ki is generated by

E(i)Rn±b2+i/2c, E(i)Rn±b3+i/2c, . . . , E(i)Rn±(n−1), and E(i)R0, (2.4.6)

where R0 stands for the longest reflection of W (A2n−1), that is,

(R1R2n−1)(R2R2n−2) · · · (Rn−1Rn+1)Rn(Rn−1Rn+1) · · · (R2R2n−2)(R1R2n−1).

The definition of yi from (2.3.1) shows that φ(yn) = R0.We will now prove that the σ-fixed part of Ki is contained in the image of φ, making

use of the fact that Ki is a Coxeter group of which the above generators are a Coxetersystem, as given by Lemma 1.5.8.

Lemma 2.4.11. Let i ∈ {0, . . . , n}. The set (2.4.6) of simple reflections of Ki is invariantunder σ. In fact, σ induces the nontrivial automorphism on the Coxeter type A2(n− j)−1 of Ki ,where j = 1+bi/2c. As a consequence, the subgroup of σ-fixed elements of Ki is of type Cn− j

and is generated by the images under φ of b(i)r j+1, b(i)r j+2, . . . , b(i)rn−1, and b(i) yn b(i).

Proof. Clearly, σ fixes E(i) = φ(b(i)). Moreover, it fixes R0 and interchanges Rn−k andRn+k, so indeed the Coxeter system of Ki is σ-invariant and σ induces the nontrivialautomorphism on the Coxeter type A2(n− j)−1 of Ki . It is well known (cf. [33]) that thesubgroup of σ-fixed elements of Ki is generated by the Coxeter system

E(i)R j+1R2n− j−1, E(i)R j+2R2n− j−2, . . . , E(i)R1R2n−1, and E(i)R0

of type Cn− j . These generators coincide with the φ-images of the simple reflections inthe statement of the lemma.

The case i = 0 of the above lemma confirms that the restriction of φ to W (Cn) is anembedding of this group into W (A2n−1) whose image coincides with the σ-fixed elementsof W (A2n−1).

The W (A2n−1)-orbit of Bi,p contains Bi , but, for p < i, these two admissible sets are indistinct W (Cn)-orbits: For B ∈B , the numbers i, the size of B, and p, the number of rootsin B fixed by σ, are constant on the W (Cn)-orbit of B in Aσ. They actually determinethis orbit uniquely.

Proposition 2.4.12. Let B ∈ Aσ be such that the number of σ-fixed roots in B is equal top and such that B has cardinality i. Then there exists an element w of the subgroup W (Cn)of W (A2n−1) such that wBi,p = B.

36 TYPE Cn

Proof. By Theorem 1.5.10 and the case i = 0 of Lemma 2.4.11, it suffices to find asymmetric diagram w in BrM(A2n−1) without horizontal strands, moving Bi,p to B.

For each γ ∈ B with γ 6= σ(γ), where γ = αt + αt+1 + · · ·+ αs, 1 ≤ t ≤ s ≤ 2n− 1,we draw four vertical strands in w as follows: from (t, 1) to (k, 0), from (2n+ 1− t, 1)to (2n+ 1− k, 0), from (s + 1, 1) to (k + 1,0), and from (2n− s, 1) to (2n− k, 0) with{αk,α2n−k} ⊂ Bi,p. For each γ ∈ B with γ = σ(γ), where γ = αn−t +αn−t+1 + · · ·+αn+t ,0 ≤ t ≤ n − 1, we draw two vertical strands: from (n − t, 1) to (n − k, 0), and from(n + t + 1, 1) to (n + k + 1, 0) where αn−k + αn−k+1 + · · · + αn+k ∈ Bi,p. Between theremaining 2n− 2i dots at the top and 2n− 2i nodes at the bottom, we just draw verticalstrands in such a way that these strands do not cross. This provides the required diagramw.

Proposition 2.4.13. The homomorphism φ : Br(Cn)→ SBr(A2n−1) is surjective.

Proof. It suffices to prove the statement for the corresponding monoids. So, let a ∈SBr(A2n−1) be a monomial. Then a represents a unique symmetric Brauer diagram.We proceed by induction on n, the case n = 1 being trivial. It is well known thatW (A2n−1)σ = φ(W (Cn)), so every symmetric Brauer diagram without horizontal arcsrepresents an element of the image under φ. Therefore, we may assume that a has atleast one horizontal strand at the top. By the action of W (Cn), we may take this strandto correspond to either {αn}= B1 or {αn−1,αn+1}. Similarly for the bottom.

For t ∈ {1,2}, view SBr(A2n−2t−1) as the subalgebra of SBr(A2n−1) spanned by allsymmetric diagrams with strands between (k, 1) and (k, 0) for k ∈ {n+ 1− t, n+ t}.

If αn occurs at both top and bottom of a, then there are v, w ∈ W (Cn) such that, upto a power of δ, either a = φ(v)En bEnφ(w) == φ(ve0)bφ(e0w) with b ∈ SBr(A2n−3).

If αn−1 and αn+1 occur at the top of a, then we can use the W (Cn)-action once moreto find either {αn−1,αn+1} = p−1({β1}) or {αn,αn−1 + αn + αn+1} = B2 at the bottom,and similarly for top and bottom interchanged. As Ep−1({β1}) = φ(e1) and EB2

= φ(b(2))by Lemma 2.4.7, we have a = φ(v f )bφ(gw) with f , g ∈ {e1, b(2)} and b ∈ SBr(A2n−5).By induction, b is in the subalgebra of SBr(A2n−1) generated by {E′n, R′nφ(ei),φ(ri) | i =1, . . . , n−2t−1}, where E′n = (Rn−t · · ·Rn−1)(Rn · · ·Rn+t−1)En(Rn−t · · ·Rn−1)(Rn · · ·Rn+t−1)for some t ∈ {1,2}, so E′n = φ(zt+1). Similarly, R′n = φ(yt+1)). It follows that a belongsto φ(BrM(Cn)), as required.

2.5 Admissible sets and their orbits

We continue with the study of the Brauer monoid BrM(Cn) of type Cn acting on Aσ, thesubset of A of σ-invariant admissible sets. This leads to a normal form for elements ofBrM(Cn) to the extent that we can provide an upper bound on the rank of Br(Cn). Thebound found in Theorem 2.5.9 is instrumental in the proof at the end of this section ofthe main Theorem 2.1.1.

Lemma 2.5.1. Let i ∈ {0, . . . , n} and p ∈ {0, . . . , i} be such that q = (i− p)/2 is an integer.Then the W (Cn)-orbit of Bi,p has size n!/(p!q!(n− i)!).

Proof. By Proposition 2.4.12, the cardinality of the orbit of Bi,p under W (Cn) is equal tothe number of diagram tops with i horizontal strands of which precisely p strands are

2.5 ADMISSIBLE SETS AND THEIR ORBITS 37

fixed by σ. This number is readily seen to be�

n

p

��

n− p

2q

�

(4q− 2)(4q− 6) · · ·2 =n!

p!q!(n− i)!.

Corollary 2.5.2. The rank a2n of SBr(A2n−1) satisfies

a2n =n∑

i=0

∑

p+2q=i

n!

p!q!(n− i)!

2

2n−i (n− i)!.

Proof. If in a symmetric diagram with 2i horizontal strands, all horizontal strands arefixed, the remaining 2(n− i) vertical strands will be in one to one correspondence withthe elements of the Weyl group of type Cn−i and order 2n−i(n−i)!. Therefore the corollaryfollows from Lemma 2.5.1.

Now we proceed to describe the stabilizer in W (Cn) of Bi,p.

Definition 2.5.3. Let i ∈ {0, . . . , n} and p ∈ {0, . . . , i} be such that i − p = 2q for someq ∈ N. By Ai,p we denote the subgroup of W (Cn) generated by the following elements:

r j ( j = 0, . . . , p− 1),rp+2k−1 (k = 1, . . . , q),t0,p = yp+1rp+1 yp+1,

tk,p = rp+2k rp+2k−1rp+2k+1rp+2k (k = 1, . . . , q− 1).

Furthermore, by Li we denote the subgroup of W (Cn) generated by yi+1, ri+1, . . ., rn−1.Finally, we set Ni,p = ⟨Ai,p, Li⟩ and let Di,p be a fixed set of representatives for left cosetsof Ni,p in W (Cn).

Figure 2.2 depicts B6,2 as a top and two elements of the form tk,p.It is easy to check that the generators of Ai,p and Li , and hence the whole group Ni,p

leaves Bi,p invariant. The next lemma shows that Ni,p is the full stabilizer of Bi,p in W (Cn).

Lemma 2.5.4. The subgroup of W (Cn) generated by {t j,p}q−1j=0 is isomorphic to W (Cq) and

the cardinality of Ai,p is 2i p!q!. Moreover, Li is isomorphic to W (Cn−i). Furthermore, Ni,pis the stabilizer of Bi,p in W (Cn) and isomorphic to Ai,p × Li .

Proof. Put

A= ⟨r0, r1, . . . , rp−1⟩,B = ⟨t0,p, t1,p, . . . , tq−1,p⟩,C = ⟨rp+1, rp+3, . . . , ri−1⟩.

Being a parabolic subgroup of type Cp, the group A is isomorphic to W (Cp). Since thesupports of the simple reflections involved in A lie in {0, . . . , p − 1} and those of B ∪ Clie in {p + 1, . . . , i}, each element of A commutes with each element of B ∪ C . Now weclaim that B is isomorphic to W (Cq). Ignoring the 2p vertical strands in the middle of

38 TYPE Cn

t0.2

t1,2

B6,2

Figure 2.2: The φ-images of the elements t0,2 and t1,2.

generators of B in BrM(A2n−1), and comparing them with canonical generators of W (Cq)in BrM(A2q−1) gives an easy pictorial proof of our claim.

Consider the diagrams of elements of B and C in BrM(A2n−1). Each diagram of B onlyhas non-crossing strands starting from (n− k − 1,1) and (n− k + 1,1), for k = p + 1,p+ 3, . . . , i − 1, which can never occur in nontrivial elements of C . Hence B ∩ C = {1}.For the generators of B and C , the following equations hold.

t0,p rp+2k−1 t0,p = rp+2k−1, for 1≤ k ≤ q

tk,p rp+2k−1 tk,p = rp+2k+1, for 1≤ k ≤ q

ts,p rp+2k−1 ts,p = rp+2k−1, for 1≤ s, k ≤ q, and |s− k|> 1.

Therefore the subgroup BC in W (Cn) is the semiproduct of C and B with C normal.Consider the diagrams of elements of BC and A in Br(A2n−1). Each element in BC keepsthe 2p strands in the middle invariant, but each element of A keeps the left 2n− 2p+ 2


strands invariant. Therefore A∩ BC = {1}. Thus, Ai,p = BC × A, and hence

|Ai,p|= |B||C ||A|= 2i p!q!.

The reflections yi+1, ri , ri+1, . . ., rn−1 have roots β0 + 2β1 + · · ·+ 2βi , βi+1, . . ., βn−1,respectively, which form a simple root system of type Cn−i . Therefore the subgroup Li isisomorphic to W (Cn−i).

In the Coxeter diagram A2n−1 we see that Ai,p∩ Li = 1 and all elements in Li commutewith all elements in Ai,p, so Ni,p is the direct product of Li and Ai,p. This gives

|Di,p| =|W (Cn)||Ai,p||Li |

=n!

p!q!(n− i)!.

By Lagrange’s Theorem, the cardinality of Di,p is equal to the size of the W (Cn)-orbit ofBi,p. Therefore by Lemma 2.5.1, Ni,p is the stabilizer of Bi,p in W (Cn).

The study of the stabilizer of Bi,p will now be used to rewrite products of bp,i,p′ withelements of W (Cn). The result is in Lemma 2.5.6 and needs the following special cases.

Lemma 2.5.5. Let i ∈ {0, . . . , n} and p ∈ {0, . . . , i} with i− p even.

(i) For each r ∈ Ai,p we have r bp,i,i = bp,i,i .

(ii) For each v ∈ Li we have vei,p = ei,p v and vbp,i,i = bp,i,i v.

Proof. (ii). By Lemma 2.3.1 and Definition 1.6.1, the two equations hold for the genera-tors of Li . Therefore they hold for each element of Li . (i). The roots β j and r j r j−1 · · · r1β0are as in Proposition 2.4.6(iii), so

r jz jz j+1 = (r jz j r jz j)r j(1.6.12)= z j(e jz j r j)

(1.6.17)= z je jz j

(1.6.29)= z j r jz j r j = z jz j+1.

This proves that (i) is satisfied with r = r j for j = 0, . . . , p−1. For the choices r = rp+2k−1for k = 1, . . . , q this is straightforward. Moreover, t0,pep+1 = yp+1(rp+1 yp+1ep+1) =yp+1 yp+1ep+1 = ep+1, and

tk,pep+2k−1ep+2k+1 = rp+2k rp+2k+1rp+2k−1rp+2kep+2k−1ep+2k+1

(1.6.8)= (rp+2k rp+2k+1ep+2k)ep+2k−1ep+2k+1

(1.6.8)+(1.6.6)= (ep+2k+1ep+2kep+2k+1)ep+2k−1

(1.5.5)= ep+2k−1ep+2k+1.

So (i) holds for all generators of Ai,p and hence for all of Ai,p.

Lemma 2.5.6. Suppose r ∈W (Cn). Let i ∈ {0, . . . , n} and p ∈ {0, . . . , i} with i− p even.

(i) There are u ∈ Di,p and v ∈ Li such that r bp,i,i = ubp,i,i v.

(ii) There are u′ ∈ Dopi,p and v′ ∈ Li such that bi,i,p r = v′bi,i,pu′.

40 TYPE Cn

Proof. Let r ∈W (Cn). By Lemma 2.5.4 and Definition 2.5.3 for Di,p, there exist u ∈ Di,p,v ∈ Li , and a ∈ Ai,p such that r = uva. By Lemma 2.5.5,

r bp,i,i = uvabp,i,i = uvbp,i,i = ubp,i,i v.

The second statement follows by applying Proposition 1.6.3 to (i).

Our next step towards a normal form for elements of BrM(Cn) is to describe productsof elements from W (Cn)bp,i,p′W (Cn) with generators e j . To this end we first prove twouseful equalities.

Lemma 2.5.7. In Br(Cn), the following hold for i ∈ {2, . . . , n− 1}.

eizi+1 = eizi , (2.5.1)

ei−1zi+1zizi−1 = ri ri−1eizizi+1zi−1, (2.5.2)

eizi+1ziei = δ2ei . (2.5.3)

Proof. By Lemma 2.4.6(iii), eizi+1(2.3.2)= ei rizi ri

(1.6.2)= eizi ri

(1.6.17)= eizi . This proves

(2.5.1).Now (2.5.3) follows from Lemma 2.4.6(i) as

eizi+1ziei(2.5.1)= eiz

2i ei = δeiziei

(1.6.15)= δ2ei .

As for (2.5.2), note that (1.6.12) and Proposition 2.4.6(iii) give zi−1ei−1zi−1 = ri−1zi−1ri−1zi−1 =zizi−1. Hence

ri−1riei−1zi+1zizi−1(1.6.8)= eiei−1zi+1zizi−1

2.3.1= eizi+1ei−1zizi−1

(2.5.1)= δeiziei−1zi−1

(2.5.1)+1.6.3= δeizi−1ei−1zi−1 = δeizizi−1

(2.5.1)= eizi+1zizi−1,

and the equation follows by left multiplication with ri ri−1.

The detailed information stated in the last sentence of the following proposition willbe needed for the proof of cellularity of Br(Cn, R,δ) in the next section.

Proposition 2.5.8. Let i, p, p′ be natural numbers with 0 ≤ p, p′ ≤ i ≤ n and i − p andi − p′ even. For each root β ∈ Ψ+, there are h ∈ {i, i + 1, i + 2}, k, m, m′ ∈ N, u ∈ Dh,m,w ∈ Dop

h,m′ , and v ∈ Lh such that

eβ bp,i,p′ = δkubm,h,m′ vw.

Moreover, if h= i, then w = 1 and m′ = p′, while k, u, and v do not depend on p′.

Proof. We first sketch the general idea of proof. There are only two possible root lengthsin the Coxeter root system Ψ of type Cn. We call β ∈ Ψ short if (β ,β) = 1 and longotherwise, in which case (β ,β) = 2. In each case, only one root needs to be considered,for all other roots of the same length are conjugate to this particular one under the naturalaction of W (Cn), and Lemma 2.5.6 can be applied to reduce to the representative root.

The top of bp,i,p′ is the admissible set Bi,p displayed in Figure 2.1.


n n+1 n+p n+i 2n1 n-i+1 n-p+1

Figure 2.3: Horizontal strands representing the three cases for a long root β .

First suppose β is long. Then β can be written as β0 + 2β1 + · · ·+ 2βt−1, for some1 ≤ t ≤ n, and so eβ = zt . We will distinguish cases according to relations among t, i,and p, and apply induction on t and i. In Figure 2.3 the roots of the Cases L1, L2, andL3 are displayed as the top, middle, and bottom horizontal strand, respectively.

Case L1. Suppose t ≥ i + 1. For any t > i + 1, we have zt = szi+1s−1, where s =rt−1 · · · ri+1. This implies that zt bp,i,p′ = szi+1 bp,i,p′s

−1, with s ∈ Li . Lemmas 2.5.6 and2.5.5 can be used to reduce this case to the case where t = i+ 1.

If p = p′ = i, then ei,p = ei,p′ = 1 and zi+1 b(i) = b(i+1) by the definition of b(i) in(2.4.1), as required.

If p 6= i = p′, it suffices to prove the equation

zi+1 bp,i,i = r bp+1,i+1,i+1,

with r in the subgroup of W (Cn) generated by r0, r1, . . . , ri . We proceed by induction oni. In view of Lemma 2.5.7, (below IH is short for Inductive Hypothesis),

zi+1 bp,i,i(2.4.1)+(2.4.4)= zi+1ei−1ei−2,pzizi−1 bi−2

2.3.1= ei−1zi+1zizi−1ei−2,p b(i−2)

(2.5.2)= ri ri−1eizizi+1zi−1ei−2,p b(i−2) 2.3.1

= ri ri−1ei(zi−1 bp,i−2,i−2)zizi+1

IH= ri ri−1ei g bp+1,i−1,i−1zizi+1 = ri ri−1 gei bp+1,i−1,i−1zizi+1

= ri ri−1 g bp+1,i+1,i+1,

where g is an element of the subgroup of W (Cn) generated by r0, r1, . . . , ri−2. Hence theclaim holds.

The case p = i 6= p′ now follows by use of Proposition 1.6.3.If p, p′ 6= i then, by the above,

zi+1 bp,i,p′ = rei+1,p+1 bi+1ei,p′ = rei+1,p+1 b(i)ei,p′zi+1 = rei+1,p+1 b(i+1)ei+1,p′+1r ′

= r bp+1,i+1,p′+1r ′,

where r, r ′ ∈ W (Cn). By Lemma 2.5.5, we conclude that this expression can be writtenin the required form with h= i+ 1.

Case L2. Next suppose p+ 1≤ t < i+ 1. By definition of zi ,

zi−1 = ri−1zi ri−1,

zi−2 = ri−2ri−3ri−1ri−2zi ri−2ri−3ri−1ri−2,

42 TYPE Cn

with ri−1, ri−2ri−3ri−1ri−2 ∈ Ai,p. By induction on t, we can find r ∈ Ai,p such thatzt = rzi r

−1. By Lemma 2.5.5,

eβ bp,i,p′ = zt bp,i,p′ = rzi r−1 bp,i,p′ = rzi bp,i,p′ .

In view of Lemma 2.5.6, this reduces the problem to rewriting zi bp,i,p′ in the requiredform.

Now

ziei−1zizi−1 = (ri−1zi−1ri−1ei−1)zi−1zi(1.6.2)+(1.6.25)

= zi−1ei−1zi−1zi

(1.6.12)= (ri−1zi−1ri−1)zi−1zi = zizi−1zi

= δzizi−1,

so

zi bp,i,i(2.4.4)= ziei−1ei−2,pzi−1zi b

(i−2) 2.3.1= ziei−1zi−1ziei−2,p b(i−2)

= δzi−1zi bp,i−2,i−2,

by the claim of Case L1 and Lemma 2.5.6, the above can be written as ubp+2,i,i v withu ∈ Di,p+2 and v ∈ Li , and so the proposition holds in this case as zi bp,i,p′ = ubp+2,i,p′ v.

Case L3. We remain with the case where 1≤ t ≤ p. We have

zt bp,i,p′ = ei,pzt b(i)ei,p′ = δei,p b(i)ei,p′ = δbp,i,p′ ,

and so the proposition holds with h= i.We next consider the case where β is a short root, which means β = βs+βs+1+· · ·+βt

with 0 < s ≤ t ≤ n − 1 or β = β0 + 2β1 + · · · + 2βs−1 + βs + βs+1 + · · · + βt with0 ≤ s ≤ t ≤ n− 1. We will distinguish seven cases by values of s and t correspondingto the horizontal strands of Figure 2.4. The seven cases occur in the order from top tobottom.

Case S1. Suppose that β is a linear combination of β j ( j = i + 1, . . . , n− 1). Then, byLemma 2.5.7,

ei+1 bp,i,p′ = ei,pei+1 b(i)ei,p′(2.5.3)= δ−2ei,p(ei+1zi+2zi+1ei+1)b

(i)ei,p′

= δ−2ei,pei+1zi+2zi+1 b(i)ei+1ei,p′

(2.4.1)+(2.4.4)= δ−2ei+2,p b(i+2)ei+2,p′ = δ

−2 bp,i+2,p′ .

At the same time, for any such β ∈Ψ+, there exists an element r ∈ Li such that β = rβi+1,thus eβ = rei+1r−1. Now eβ bp,i,p′ = δ−2r bp,i+2,p′ r

−1, and hence the proposition holdswith h= i+ 2.

Case S2. Suppose β = βs + βs+1 + · · ·+ βt , with p ≤ s ≤ i ≤ t ≤ n− 1.If p = i, then ei,p = 1. First, consider the case t = s = i. Since

eizi+1zi(2.5.1)= eizizi

2.4.6(i)= δeizi ,


1 n-i+1n-p+1

n+p n+i 2nSubcase 1

Subcase 2

Subcase 3

Subcase 4

Subcase 5

Subcase 6

Subcase 7

Figure 2.4: Strands corresponding to 7 possibilities for the long root β .

44 TYPE Cn

the use of the claim of Case L1 and Proposition 1.6.3 gives the existence of an elementr ∈W (Cn) such that

ei bi,i,p′ = δ−1eizi b(i)ei,p′

2.4.6(i)= δ−1eizi+1 b(i)ei,p′

2.3.1= δ−1ei bi,i,p′zi+1

L1= δ−1ei bi+1,i+1,p′ r = δ

−1 bi−1,i+1,p′ r,

as required.If t 6= s, then eβ = rβ ′′ rβ ′ ei rβ ′ rβ ′′ , where β ′ = βs + · · ·+ βi−1, β ′′ = βi+1 + · · ·+ βt ,

which implies that rβ ′ ∈ Ai,p, rβ ′′ ∈ Li , and hence eβ b(i)ei,p′ = rβ ′′ rβ ′ ei b(i)ei,p′ rβ ′′ . As in

the argument for ei bi,i,p′ above, this can be written in the required form with h= i+ 1.On the other hand if p 6= i , then ei,p 6= 1. Therefore for β = βs + βs+1 + · · ·+ βt ,

with p ≤ s ≤ i ≤ t ≤ n− 1, since there is some l ∈ {p + 1, p + 3, . . . , i − 1}, such thateβ el = rl rβ el , implying that eβ ei,p = rl rβ ei,p and eβ bp,i,p′ = rl rβ bp,i,p′ . By Lemma 2.5.6,the proposition holds with h= i.

Case S3. Suppose β = βs + βs+1 + · · ·+ βt or β = β0 + 2β1 + · · ·+ 2βs−1 + βs + · · ·+ βtwith 0< s ≤ p and i ≤ t ≤ n− 1.

First consider β = βp + · · ·+ βi . Following the argument for eizi+1zi = δeizi in theabove case, we find that eβzi+1zi = δeβzi holds, which implies

eβ bp,i,p′ = ei,peβ b(i)ei,p′ = δ−1ei,peβ bi+1ei,p′ . (2.5.4)

Observe that

ri−1rieβ ei−1 = eβ−βi−βi−1ri−1riei−1ri ri−1ri−1ri

(1.6.9)= eβ−βi−βi−1

ei ri−1ri ,

and ri−1ri b(i+1) 2.5.5

= b(i+1).

Therefore (2.5.4) can be written as

δ−1ei,peβ bi+1ei,p′ = δ−1(ei−1ri ri−1)ei−2,peβ−βi−βi−1b(i+1)ei,p′

(1.6.9)= δ−1ri ri−1eiei−2,peβ−βi−βi−1

b(i+1)ei,p′

= δ−1ri ri−1ei(eβ−βi−βi−1ei−2,p b(i−2))zi−1zizi+1ei,p′ .

By induction on i, we can use an argument as in the claim of Case S1, and the above canbe written as

δ−1ri ri−1ei gei−1,p−1 b(i−1)zi−1zizi+1ei,p′ = ri ri−1 geiei−1,p−1 b(i−1)zizi+1ei,p′

= ri ri−1 gei+1,p−1 b(i)zi+1ei,p′

= ri ri−1 gei+1,p−1 bi,i,p′zi+1,

where g ∈W (Cn) is a product of elements from r0, r1, . . . , ri−2. By Case L1 and Proposi-tion 1.6.3, the proposition holds with h= i+ 1.

We return to the general setting of Case S3. Then there exists r ′ ∈ Li and r ′′ ∈ Ai,p

such that r ′r ′′β = β , with β = βp + · · ·+ βi . Then

eβ bp,i,p′ = r ′r ′′eβ bp,i,p′ r′,


hence the proposition holds in Case S3 due to Lemma 2.5.6.

Case S4. If β = β0+ 2β1+ · · ·+ 2βs−1+βs + · · ·+βt with p ≤ s ≤ i ≤ t ≤ n− 1, and letβ ′ = βs + · · ·+ βt . Then eβ = yseβ ′ ys. When p = i = s, we see that ys ∈ Ai,p; when p 6= i,there is some l ∈ {p+ 1, p+ 3, . . . , i − 1}, such that eβ el = rl rβ el . This brings us back tothe argument of Case S2.

Case S5. If β = β0 + 2β1 + · · · + 2βs + βs+1 + · · · + βt , with i < s ≤ t ≤ n − 1, thenβ = ys+1β

′, with β ′ = βs+1 + βs+2 + · · ·+ βt , and ys+1 ∈ Li . Therefore

eβ bp,i,p′ = ys+1(eβ ′ bp,i,p′)ys+1,

and we are back in Case S1.

Case S6. If β = β0 + 2β1 + · · ·+ 2βs + βs+1 + · · ·+ βt or β = βs + βs+1 + · · ·+ βt , with0 ≤ s ≤ t and p ≤ t ≤ i − 1, then there must be some e j ∈ {ep+2 j−1}

(i−p)/2j=1 such that β

is not orthogonal to β j , or {β ,β j} is not admissible, or β = β j . Then, by Lemma 2.4.6,there exists r ∈ W (Cn) such that eβ ei,p = rei,p or eβ ei,p = δrei,p. This implies that theproposition holds with h= i.

Case S7. If β can be written as a linear combination of {β j}p−1j=0 , then β is conjugate to

βp−1 under the subgroup of Ai,p generated by {r j}p−1j=0 . Then we can find a r ∈ Ai,p such

that rβp−1 = β , which implies

eβ bp,i,p′ = rep−1r−1 bi,i,p′2.5.5(i)= rep−1 bi,i,p′ = r bp−2,i,p′ ,

so the proposition holds with h= i due to Lemma 2.5.6.

Theorem 2.5.9. Each element in the monoid BrM(Cn) can be written as

δkubp,i,p′ vwop,

where k ∈ Z and i, p, p′ ∈ {0, . . . , n} with i − p and i − p′ even, u ∈ Di,p, v ∈ Li , andw ∈ Di,p′ . In particular, Br(Cn) is free of rank at most a2n.

Proof. Let U be the set of elements of BrM(Cn) of the indicated form. We show that Uis invariant under left multiplication by generators of BrM(Cn). To this end, consideran arbitrary element a = δkubp,i,p′ vwop of U . Obviously δ±1a ∈ U . Without loss ofgenerality, we may take k = 0.

Let r ∈ W (Cn). By Lemma 2.5.6 applied to ru there are u′ ∈ Di,p and v′ ∈ Lisuch that ra = rubp,i,iei,p′ vwop = u′bp,i,i v

′ei,p′ vwop. By Lemma 2.5.5(ii), this is equalto u′bp,i,iei,p′ v

′vwop = u′bp,i,p′ v′vwop and, as v′v ∈ Li , the set U is invariant under left

multiplication by Weyl group elements.Finally, consider the generator e j of BrM(Cn). Writing β = uopα j we have e ja =

e jubp,i,p′ vwop = ueβ bp,i,p′ vwop and by Proposition 2.5.8 this belongs to U again.Now, by Proposition 1.6.3 we also find that U is invariant under right multiplication

by generators. This proves that U is invariant under both left and right multiplication byany generator of BrM(Cn). As it contains the identity (b0,0,0), it follows that U coincideswith the whole monoid.

As for the last assertion of the theorem, observe that freeness of Br(Cn) over Z[δ±1] isimmediate from the fact that it is a monomial algebra with a finite number of generators.

46 TYPE Cn

By the first assertion, its rank is at most

n∑

i=0

∑

p≡i (mod 2)

|Di,p|

2

· |Li |.

By Lemma 2.5.1, the cardinality of Di,p is n!/(p!q!(n− i)!), where q = (i− p)/2, and, byLemma 2.5.4, Li is isomorphic to W (Cn−i), which has 2n−i(n− i)! elements. Therefore,the rank of Br(Cn) over Z[δ±1] is at most

∑

i

∑

p,q:p+2q=i

n!

p!q!(n− i)!

2

2n−i(n− i)!.

Corollary 2.5.2 gives that this sum is equal to a2n.

We are now ready to prove Theorem 2.1.1. Theorem 2.5.9 shows that Br(Cn) is freeof rank at most a2n. By Proposition 2.4.13, the homomorphism φ : Br(Cn)→ SBr(A2n−1)is surjective, so the rank of Br(Cn) is at least the rank of SBr(A2n−1), which is known tobe a2n by Corollary 2.2.3. Thus, the ranks of Br(Cn) and SBr(A2n−1) coincide and φ is anisomorphism.

2.6 Further properties of type C algebras

In this section we prove that the algebra BrM(Cn, R,δ) is cellular, in the sense of Grahamand Lehrer [26], provided R is an integral domain containing the inverse to 2. Theproof given here runs parallel to the proof of the corresponding result for Dn in [11,Section 6]. Thanks to [29], an alternative proof is possible, which consists of showingthat Br(Cn, R,δ) is a cellularly stratified diagram algebra; this implies that Br(Cn, R,δ) iscellular if the group algebra of W (Cn) over R is cellular. This approach is described byBowman [2] on the basis of a preliminary version of our paper. We finish this section bydiscussing a few more desirable properties of the newly found Brauer algebras.

For ∗ we will use the anti-involution op determined in Proposition 1.6.3. Let i ∈{0, . . . , n}. By Theorem 2.5.9, each element in the monoid BrM(Cn) can be written in theform

δkubp,i,p′ vwop,

where k ∈ Z and i, p, p′ ∈ {0, . . . , n} are such that i − p and i − p′ are even, u ∈ Di,p,v ∈ Li , and w ∈ Di,p′ . As the coefficient ring R is an integral domain containing theinverse of 2, it satisfies the conditions of [24, Theorem 1.1], so by [24, Corollary 3.2] thegroup rings R[Li] (i = 0, . . . , n) are all cellular. By Lemma 2.5.4, the subalgebra RLi ofBr(Cn, R,δ) (with unit b(i) of Definition 2.4.10) generated by Li is isomorphic to R[Li].Let (Λi , Ti , Ci ,∗i) be a cell datum for RLi an in [24]. Observe that the generators of Li inDefinition 2.5.3 are fixed by op, so RLi is op-invariant. By [24, Section 3], ∗i is the mapop on RLi and so ∗i is the restriction of op to RLi .

The underlying poset Λ will be {Bi}ni=0, as defined in (2.4.2). We say that Bi > B j ifand only if i < j or, equivalently, Bi ⊂ B j . In particular, ; is the greatest element of Λ.

2.6 FURTHER PROPERTIES OF TYPE C ALGEBRAS 47

The set T (Bi) is taken to be the set of all triples (u, ei,p, s) where u ∈ Di,p (see Defi-nition 2.5.3), p ∈ {0, . . . , i} with i − p even, and the product ei,p is given by (2.4.4), ands ∈ Ti . Clearly, this set is finite.

The map C is given by C((u, ei,p, s), (w, ei,p′ , t)) = uei,pCi(s, t)ei,p′wop. By Lemma

2.5.5, ubp,i,p′ vwop = (uei,p)(b(i)v)(wei,p′)op, so the image of C is a basis by Theorems2.1.1 and 2.5.9, and the fact that {Ci(s, t) | s, t ∈ Ti} is a basis for RLi (which is aconsequence of (C1) for (Λi , Ti , Ci ,∗i)). This gives a quadruple (Λ, T, C ,∗) satisfying(C1).

For (C2) notice that (uei,pCi(s, t)ei,p′wop)op = wei,p′Ci(s, t)opei,puop. Now Ci(s, t)op =

Ci(t, s), by the cellularity condition (C2) for RLi and so (C2) holds for the cell datum(Λ, T, C ,∗).

Finally, we check condition (C3) for (Λ, T, C ,∗). It suffices to consider the left multi-plications by r j and e j of uei,pCi(s, t)ei,p′w

op. Up to linear combinations, we can replacethe latter expression by uei,p b(i)vei,p′w

op for v ∈ Li . Now, by Lemma 2.5.6 for r j andProposition 2.5.8 for e j (notice that the product lies in Br(Cn, R,δ)<Bi

if h > i), (C3)holds for the cell datum (Λ, T, C ,∗). Therefore we have now proved

Theorem 2.6.1. Let R be an integral domain with 2−1 ∈ R. Then the quadruple (Λ, T, C ,∗)is a cell datum for Br(Cn, R,δ), proving the algebra is cellular.

The reader may have wondered why the study of symmetric diagrams was restrictedto type Am for m odd. The answer is that, if m = 2n is even, each symmetric diagramhas a fixed vertical strand from the dot (n, 1) to (n, 0). The removal of this strand leadsto an isomorphism of the algebra of the symmetric diagrams with SBr(A2n−1), and sothis construction provides no new algebra. This is remarkable in that the root systemobtained by projecting Φ onto the σ-fixed subspace of the reflection representation, as inDefinition 2.4.1, leads to a root system of type Bn instead of Cn.

At the time of writing of the paper [14], Chen [8] presented a definition of a gener-alized Brauer algebra of type I2(m). For m = 4, this type coincides with C2, but Chen’salgebra has dimension 2m+m2 = 24, whereas our Br(C2) has dimension 25.

48 TYPE Cn

3TYPE Bn

This chapter is based on our paper [15].

3.1 Introduction

In [9], the Brauer algebra Br(Q) of any simply-laced Coxeter type was defined in such away that for Q = An−1, the classical Brauer algebra of diagrams on 2n nodes emerges.For these algebras, a deformation to a Birman–Murakami–Wenzl (BMW) algebra wasdefined and in [10], and, for the spherical types among these, the algebra structure wasfully determined in [12, 17]. Again, for Q = An−1, the classical BMW algebras re-appear.

The starting point for an extension of these algebras to non-simply laced diagrams,begun in Chapter 2, is based on the following observation. It is well known that the Cox-eter group of type Bn arises from the Coxeter group of type Dn+1 as the subgroup of allelements fixed by the nontrivial Coxeter diagram automorphism. Crisp [19] showed thatthe Artin group of type Bn arises in a similar fashion from the Artin group of type Dn+1. Inthis chapter, we study the subalgebra SBr(Dn+1) of the Brauer algebra Br(Dn+1) spannedby the monomials fixed under the automorphism induced by the nontrivial Coxeter dia-gram automorphism. We also give a presentation of this subalgebra by generators andrelations, which we regard as the definition of a Brauer algebra of type Bn. This chaptercontinues Chapter 2 of a Brauer algebra of type Cn of Br(A2n−1) spanned by monomialsfixed under the canonical Coxeter diagram automorphism.

Each defining relation (given in Definition 1.6.1 below) concerns at most two indices,say i and j, and is (up to the parameters in the idempotent relation) determined by thediagram induced by Bn on {i, j}. The nodes of the Dynkin type Bn are labeled as follows.

Bn = ◦n−1

◦n−2· · · · · · ◦

2◦1> ◦

0.

The generators of the Brauer algebra Br(Bn) are denoted r0,. . ., rn−1, e0,. . ., en−1. In orderto distinguish these from the canonical generators of the Brauer algebra of type Dn+1, thelatter are denoted R1, . . . , Rn+1, E1, . . . , En+1 instead of the usual lower case letters (see

50 TYPE Bn

Definition 1.3.1). The diagram for Dn+1 is depicted below.

Dn+1 = ◦n+1

◦n· · · · · · ◦

4

2◦

◦3

◦1

.

Definition 3.1.1. Let σ be the natural isomorphism on BrM(Dn+1) (Definition 1.3.1),which is induced by the action of the permutation (1, 2) on the indices of the genera-tors {Ri , Ei}n+1

i=1 of BrM(Dn+1) and keeping the parameter δ invariant. The fixed sub-monoid of BrM(Dn+1) under σ is called the symmetric submonoid of BrM(Dn+1), anddenoted SBrM(Dn+1). The linear span of SBrM(Dn+1) is called the symmetric subalgebraof Br(Dn+1), and denoted SBr(Dn+1).

Theorem 3.1.2. There exists a Z[δ±1]-algebra isomorphism

φ : Br(Bn)−→ SBr(Dn+1)

determined by φ(r0) = R1R2, φ(ri) = Ri+2, φ(e0) = E1E2, and φ(ei) = Ei+2, for 0 < i ≤n− 1. Furthermore both algebras are free of rank

f (n) := 2n+1 · n!!− 2n · n!+ (n+ 1)!!− (n+ 1)!.

In Theorem 3.5.1 of this chapter, we also show that these algebras are cellular in thesense of Graham and Lehrer [26]. The subalgebra of Br(Bn) generated by r0, . . . , rn−1 iseasily seen to be isomorphic to the group algebra of the Weyl group W (Bn) of type Bn.

Here we give the first few values of ranks of Br(Bn).

n 1 2 3 4 5f (n) 3 25 273 3801 66315

The case n = 2 is discussed in Chapter 2, as B2 and C2 represent the same diagram.Here we illustrate our results with the next interesting case: n= 3. Let

F = {1, e2, e0, e1e0, e0e1, e1e0e1, e0e2, e2r1e0e1e2}. (3.1.1)

We have the following decomposition of Br(B3) into Z[δ±1]W (B3)-submodules, whereW (B3) is the submonoid of Br(B3) generated by r0, r1, r2.

Br(B3) =⊕

e∈F

Z[δ±1]W (B3)eW (B3).

Besides, the sizes of W (B3)eW (B3) are 48, 144, 18, 18, 18, 9, 9, 9 for e ∈ F in the orderthey are listed. This accounts for the rank of Br(B3) being 273.

The strategy of proof is as follows. The monomials in the canonical generators ofBr(Dn+1) are known to correspond to certain Brauer diagrams with an additional deco-ration of order two by means of the isomorphism ψ : Br(Dn+1)→ BrD(Dn+1) introduced

3.1 INTRODUCTION 51

in [13]; here BrD(Dn+1) is an algebra linearly spanned by the decorated classical Brauerdiagrams, and the details are described in Section 3.2. The proof then consists of show-ing that the image of φ is the linear span of the symmetric diagrams, which is free ofrank f (n), and that Br(Bn) is linearly spanned by at most f (n) monomials. The latter iscarried out by means of rewriting monomials to normal forms, in such a way that eachBrauer diagram corresponds to a unique normal form. This process leads to a basis whichcan be shown to be cellular.

This chapter has five sections. Section 3.2 gives some elementary properties of Br(Bn)in preparation of Section 3.4. We also recall results of Brauer algebras of type Dn+1 andpresent the surjectivity of the map φ of Theorem 3.2.8 by combinatorial arguments inthis section. Section 3.3 discusses aspects of the root system of type Bn that are used toidentify monomials of Br(Bn) in r0, . . ., rn−1, e0, . . ., en−1; moreover a pictorial descriptionof some monomial images in BrD(Dn+1) is presented. In Section 3.4, the rewriting ofmonomials of Br(Bn) is established, which leads to an upper bound on the rank of Br(Bn)and the main theorem is proved at the end of this section. In the last section, we establishthat Br(Bn) is cellular.

The idea of obtaining non-simply laced algebras from simply laced types has beenapplied in [21] for Temperley-Lieb algebras of type B with generators ei , in [30] for thereduced BMW algebras of type B and in [25] for Hecke algebras of type B. In [8], Z. Chendefines a Brauer algebra for each pseudo-reflection group by use of a flat connection. Fortypes B and C, it is different from our algebra and has some intricate relations with ouralgebras to be explained in further research.

52 TYPE Bn

3.2 Elementary properties

Recall the definition of Br(Bn) from Definition 1.6.1. It is a direct consequence of the def-inition that the submonoid of BrM(Bn) generated by {ri}n−1

i=0 (the Weyl group generators)is isomorphic to the Weyl group W (Bn) of type Bn. The algebra Br(B2) is isomorphic toBr(C2) in Chapter 2, and the isomorphism is given by exchanging the indices 0 and 1of the Weyl group generators and of the Temperley-Lieb generators. As a consequence,Lemma 1.6.2 applies in the following sense.

Lemma 3.2.1. In Br(Bn), the following equalities hold.

e1e0e1 = e1r0e1 (3.2.1)

r0e1e0 = e1e0 (3.2.2)

e0r1r0e1 = e0e1 (3.2.3)

r1r0e1r0 = r0e1r0r1 (3.2.4)

e1r0e1r0 = e1e0e1 (3.2.5)

We recall Defintion 1.3.1 and Remark 1.5.1. In order to avoid confusion with theabove generators, the symbols of [9] for the generators of Br(Q) have been capitalized.

Since Br(B2) ∼= Br(C2) and Br(D3) ∼= Br(A3), the following corollary can be verifiedeasily as in Chapter 2.

Corollary 3.2.2. The map defined as φ on the generators of Br(Bn) in Theorem 3.1.2extends to a unique algebra homomorphism φ on Br(Bn). Furthermore, the image of φ iscontained in SBr(Dn+1).

Proof. The first claim can be verified by checking defining relations under φ. The secondclaim holds for the image of each generator of Br(Bn) under φ is in SBr(Dn+1).

Since the subalgebra in Br(Dn+1) generated by {Ri , Ei}n+1i=3 is isomorphic to Br(An−1),

which can be found in [13], the proposition below holds naturally.

Proposition 3.2.3. The subalgebra generated by {ri , ei}n−1i=1 and δ in Br(Bn) is isomorphic

to Br(An−1).

Hence the formulas in Remark 1.5.1 still hold for lower letters with nonzero indices.The next two lemmas contain formulas that will be applied in Section 3.4.

For 2≤ i ≤ n− 1, set e∗1 = r0e1r0 and e∗i = ri−1rie∗i−1ri ri−1.

Lemma 3.2.4. For i ∈ {1, . . . , n− 1},

eie∗i = eiei−1 · · · e1e0e1 · · · ei−1ei , (3.2.6)

r0eie∗i = eie

∗i . (3.2.7)

3.2 ELEMENTARY PROPERTIES 53

Proof. For i = 1, we have e1e∗1 = e1r0e1r0(3.2.5)= e1e0e1. For i > 1 induction, (1.6.9), and

(1.6.1) give ri−1riei−1ri ri−1 = ei , so

eie∗i = ri−1riei−1ri ri−1ri−1rie

∗i−1ri ri−1

(1.6.1)= ri−1riei−1e∗i−1ri ri−1

= (ri−1riei−1) · · · e1e0e1 · · · (ei−1ri ri−1)(1.6.8)+(1.5.1)= eiei−1 · · · e1e0e1 · · · ei−1ei .

This establishes (3.2.6). Equality (3.2.7) follows from (3.2.6), (1.6.5), and (3.2.2).

Put

g = e2r1e0e1e2. (3.2.8)

Lemma 3.2.5. For n≥ 3, the following equations hold in Br(Bn),

g = gop, (3.2.9)

(r1r0r1)e2r1e0e1 = e2r1e0e1, (3.2.10)

(r1r0r1)g = g, (3.2.11)

e0 g = δe0e2. (3.2.12)

For n≥ 4, the following equations hold in Br(Bn),

(r3r2r1r0r1r2r3)g = g, (3.2.13)

e0r1r2r3 g = δe0e1e3r2r3, (3.2.14)

e0r1 g = δe0r2r1e2, (3.2.15)

e1r2r3 g = e1e0r1r2r3e2. (3.2.16)

Proof. From

g = (e2r1)e0e1e2(1.6.5)= e2e1(r2e0)e1e2

(1.6.5)= e2e1e0(r2e1e2)

(1.5.2)= e2e1e0r1e2,

it follows that g is invariant under opposition, and so (3.2.9). Equality (3.2.10) followsfrom

r1r0(r1e2r1)e0e1(1.6.9)= r1(r0r2)e1(r2e0)e1

(1.6.5)+(1.6.6)= r1r2(r0e1e0)r2e1

(3.2.2)= r1r2e1(e0r2)e1

(1.6.6)= (r1r2e1r2)e0e1

(1.6.9)= e2r1e0e1.

Equality (3.2.11) follows from (3.2.10) by right multiplication by e2, and Equality (3.2.13)from

r3r2r1r0r1r2r3 g(1.6.8)= r3r2(r1r0r1e3)g

(1.6.5)= r3r2e3r1r0r1 g

(3.2.11)= r3r2e3 g

(1.6.8)+(1.5.5)= g.

Formula (3.2.12) follows from

e0 g = (e0e2)r1e0e1e2(1.6.6)= e2(e0r1e0)e1e2

(1.6.14)= δe2e0e1e2

(1.6.6)+(1.5.5)= δe0e2,

54 TYPE Bn

Formula (3.2.14) from

e0r1r2r3 g = e0r1(r2r3e2)r1e0e1e2(1.6.8)= e0(r1e3)e2r1e0e1e2

(1.6.5)= e0e3(r1e2r1)e0e1e2

(1.6.9)= (e0e3r2)e1(r2e0)e1e2

(1.6.5)+(1.6.6)= e3r2(e0e1e0)r2e1e2

(1.6.15)= δ(e3r2e0r2e1)e2

(1.6.1)+(1.6.5)+(1.6.6)= δe0e1e3e2

(1.5.1)= δe0e1e3r2r3.

Formula (3.2.15) from

e0r1 g = e0(r1e2r1)e0e1e2(1.6.9)= (e0r2)e1(r2e0)e1e2

(1.6.5)= r2(e0e1e0)r2e1e2

(1.6.15)= δ(r2e0r2)e1e2

(1.6.1)+(1.6.5)= δe0(e1e2)

(1.6.8)= δe0r2r1e2,

and Formula (3.2.16) from

e1r2r3 g = e1(r2r3e2)r1e0e1e2(1.6.8)= (e1e3)e2r1e0e1e2

(1.6.6)= e3(e1e2r1)e0e1e2

(1.5.4)= (e3e1r2e0)e1e2

(1.6.5)+(1.6.6)= e1e0e3(r2e1e2)

(1.5.2)= e1e0(e3r1)e2

(1.6.5)= e1e0r1(e3e2)

(1.6.8)= e1e0r1r2r3e2.

In order to give the diagram interpretation of monomials of Br(Bn), we recall theBrauer diagram algebra of type Dn+1 from [13]. Divide 2n + 2 points into two sets{1, 2, . . . , n + 1} and {1, 2, . . . ,Ön+ 1} of points in the (real) plane with each set on ahorizontal line and point i above i. An n+1-connector is a partition on 2n+2 points inton+1 disjoint pairs. It is indicated in the plane by a (piecewise linear) curve, called strandfrom one point of the pair to the other. A decorated n+ 1-connector is an n+ 1-connectorin which an even number of pairs are labeled 1, and all other pairs are labeled by 0. Apair labeled 1 will be called decorated. The decoration of a pair is represented by a blackdot on the corresponding strand. Denote Tn+1 the set of all decorated n+ 1-connectors.Denote T 0

n+1 the subset of Tn+1 of decorated n+ 1-connectors without decorations anddenote T=n+1 the subset of Tn+1 of decorated n+1-connectors with at least one horizontalstrand.

Definition 3.2.6. Let H be the commutative monoid with presentation

H =¬

δ±1,ξ,θ | ξ2 = δ2,ξθ = δθ ,θ 2 = δ2θ¶

=¬

δ±1¶

{1,ξ,θ}.

A Brauer diagram of type Dn+1 is the scalar multiple of a decorated n-connector by anelement of H belonging to

¬

δ±1¶

(Tn+1 ∪ ξT=n+1 ∪ θ(T0n+1 ∩ T=n+1)). The Brauer diagram

algebra of type Dn+1, denoted BrD(Dn+1), is the Z[δ±1]-linear span of all Brauer diagramsof type Dn+1 with multiplication laws defined in [13, Definition 4.4].

The scalar ξδ−1 appears in various products of n + 1-connectors described in [13,Definition 4.4] and two consecutive black dots on a strand are removed. The multipli-cation is an intricate variation of the multiplication in classical Brauer diagrams, where


��

= δ, = θ ,��u��

��u

uu = ξ

Figure 3.1: The closed loops corresponding to the generators of H

1 2 n+ 1 1 2 n+ 1

1 2 n+ 1 1 2 n+ 11 i− 1 i n+ 1 1 i− 1 i n+ 1

1 i− 1 i n+ 1 1 i− 1 i n+ 1

· · · · · ·

· · · · · ·· · · · · ·

ψ(R1) =ψ(E1) =

ψ(Ri) = ψ(Ei) =2 ≤ i ≤ n+ 1

Figure 3.2: The images of the generators of Br(Dn+1) under ψ

the points of the bottom of one connector are joined to the points of the top of the otherconnector, so as to obtain a new connector. In this process, closed strands appear whichare turned into scalars by translating them into elements of H as indicated in Figure 3.1.

In [13], the algebra BrD(Dn+1) is proved to be isomorphic to Br(Dn+1) by means ofthe isomorphism ψ : Br(Dn+1) 7→ BrD(Dn+1) defined on generators as in Figure 3.2. It isfree over Z[δ±1] with basis Tn+1 ∪ ξT=n+1 ∪ θ(T

0n+1 ∩ T=n+1).

Definition 3.2.7. Write T |n+1 for the subset of Tn+1 consisting of all n+1-connectors with

a fixed strand from 1 to 1, and set T |=n+1 = T |n+1∩T=n+1. It is readily checked that the union

of δZT |n+1 , δZξT |=n+1, and δZθ(T=n+1 ∩ T 0n+1) is a submonoid of BrD(Dn+1); we denote it

by BrMD(Bn) and the corresponding algebra over Z[δ±1] by BrD(Bn).

The images of the generators of Br(Bn) under ψφ in BrD(Dn+1) lie in BrD(Bn); theyare indicated in Figure 3.3.

Theorem 3.2.8. For φ and ψ, the following holds.

(i) The restriction of ψ to φ(Br(Bn)) is an isomorphism onto BrD(Bn).

(ii) The image of φ coincides with SBr(Dn+1).

(iii) The Z[δ±1]-algebras SBr(Dn+1) and BrD(Bn) are free of rank f (n).

These assertions imply the commutativity of the following diagram.

56 TYPE Bn

ψφ(r0) =

1 2

ψφ(ri) =

i+ 1 i+ 2

1 2

n+ 1

n+ 1

1

1 i+ 1 i+ 2

n+ 1

n+ 1

ψφ(e0) = θδ

1 2

ψφ(ei) =

i+ 1 i+ 2

3 n+ 1

1 2 3 n+ 1

1 n+ 1

1 i+ 1 i+ 2 n+ 1

Figure 3.3: The images under ψφ of the generators of Br(Bn)

Br(Bn)φ

%%

// SBr(Dn+1)� _

��

∼=// BrD(Bn)� _

��Br(Dn+1)

ψ

∼=// BrD(Dn+1)

Proof. (i). All of the images of the generators {ri , ei}n−1i=0 under ψφ are in BrD(Bn).

Therefore, the assertion is equivalent to the statement that all elements in BrMD(Bn) canbe written as products of {ψφ(ri), ψφ(ei)}n−1

i=0 up to some powers of δ.Before we start to verify this fact, we introduce three kinds of special element Ki ,

Eεi+ε j, and and Eεi−ε j

in BrD(Bn). The first two are as indicated in Figure 3.4. The thirdis as Eεi+ε j

, but without the decoration. These are all in ψφ(Br(Bn)), as can be verifiedby induction on i and j (alternatively, they are ψφ(rεi

), ψφ(eεi+ε j), and ψφ(eεi−ε j

) inthe notation of Section 3.3).

Let a ∈ BrMD(Bn). We will show a ∈ψφ(Br(Bn)). For n= 2, 3, the required assertioncan be checked by use of diagrams. We proceed by induction on n and let n> 3.


Ki = Eεi+εj =

1 i n+ 1 1 i j n+ 1

1 i n+ 1 1 i j n+ 1

· · · · · · · · · · · · · · ·

2 ≤ i ≤ n+ 1 2 ≤ i < j ≤ n+ 1

Figure 3.4: Ki and Eεi+ε j

.

First assume a ∈ δZT |n+1 ∪ δZξT |=n+1. Notice that a has a vertical strand from 1 to

1. If a has another vertical strand, say between i and j with 1 < i, j ≤ n+ 1, then, bymultiplying a by a suitable element of

ψφ(ri)�n−1

i=1 (which is isomorphic to W (An−1))at the left and by another element of it at the right, we can move this vertical strand sothat it connects n+1 and Ön+ 1. If it is decorated, then we apply Kn+1 (an element in theimage ofψφ) to remove the decoration on this strand; as a consequence, we may restrictourselves to a ∈ BrD(Bn−1). But then induction applies and gives a ∈ψφ(Br(Bn)).

If a has only one vertical strand, then the strands from n+1 and Ön+ 1 are horizontal.As above, we multiply by two suitable elements of the monoid generated by

�

ψφ(ri)�n−1

i=1 ,which is isomorphic to W (An−1), to move the top horizontal strand to the strand connect-ing n and n+1 and the bottom horizontal strand to one connecting n and Ön+ 1. Next weapply Kn+1 to remove possible decorations on the two strands. As a result, a ∈ Br(Bn−2)and so a ∈ψφ(Br(Bn)) by the induction hypothesis.

The case a ∈ δZθ(T=n+1 ∩ T 0n+1) remains. We distinguish five possible cases for a by

the strands with ends 1 and 1.

• M (2) is the subset of T=n+1∩T 0n+1 of all diagrams with a fixed vertical strand between

1 and 1,

• M (3) is the subset of T=n+1 ∩ T 0n+1 of all diagrams with two different horizontal

strands with ends 1 and 1,

• M (4) is the subset of T=n+1 ∩ T 0n+1 of all diagrams with two different vertical strands

with ends 1 and 1,

• M (5) is the subset of T=n+1 ∩ T 0n+1 of all diagrams with a horizontal strand starting

from 1 and a vertical strand from 1, and

• M (6) is the subset of T=n+1 ∩ T 0n+1 of all diagrams with a horizontal strand starting

from 1 and vertical strands from 1.

If a ∈ δZθM (2), the diagram part can be written as a scalar multiple of the imageof some monomial b ∈ δZ

ri , ei�n−1

i=1 , so a = δkθψφ(b) for some k ∈ Z. As a ∈ T=n+1,there are i, j ∈ {1, . . . , n+ 1} with i < j such that a has a horizontal strand at the topbetween i and j, where 1 < i < j ≤ n+ 1. Now a = δk−1θ Eεi−ε j

ψφ(b) for some k ∈ Z.

58 TYPE Bn

As θδ−1Eεi−ε j= Eεi−ε j

Eεi+ε j(see Definition 4.4(v) of [13]) and both Eεi−ε j

and Eεi+ε j

belong to the image of ψφ, we are done.If a ∈ δZθM (i) for i = 5, 6, 4, 3, respectively, then, by multiplying by suitable ele-

ments inψφ

ri�n−1

i=1 (which is isomorphic to W (An−1)) at both sides of a, we can achievethat the strands between {i, i}mi=1 for m = 3, 3,4, 2 are as in the diagram of ψφ(b) forb = e0e1, e1e0, g, e0, respectively. Up to the leftmost m strands, the resulting diagramcan be considered as an element of δZψφ

ri , ei�n−1

i=m, so a ∈ δZθψφ(Br(Bn)). As before,there are i, j ∈ {1, . . . , n+1} with i < j and k ∈ Z such that a = δk Ei, jψφ(Bn)⊆ψφ(Bn),and so Assertion (i) holds.

(ii). It follows from (i) and Corollary 3.2.2 that

ψ−1(BrD(Bn)) = φ(Br(Bn))⊆ SBr(Dn+1).

Therefore it suffices to prove that SBrM(Dn+1) is contained in φ(BrM(Bn)), or, equiva-lently, with BrMD(Dn+1) as in Definition 3.2.6,

�

BrMD(Dn+1) \ BrMD(Bn)�

∩ψ(SBrM(Dn+1)) = ;.

We find that BrMD(Dn+1)\BrMD(Bn) consists of all diagrams belonging to one of δZM (i)∪ξδZM (i), for i = 3, . . . , 6, adorned with an even decoration. By an argument analogous tothe above and subsequently multiplying by Ki to remove decorations on all strands exceptthe leftmost 3 or 4 strands, we can reduce the verification to a case where n = 2,3, andso finish by induction.

For example, for n = 5, the diagram at the left in Figure 3.5 belongs to M (4). Bymultiplying by the elements r5r3r4 and r3r4r5 of ⟨r3, r4, r5, r6⟩ (which is isomorphic toW (A4)) at the top (left of the corresponding monomial) and the bottom (right of thecorresponding monomial), respectively, we obtain the diagram at the right of the figure.We find that the part in BrMD(D4) is R1E4 and σ(R1E4) = R2E4 6= R1E4; hence it is not inψ(SBrM(D6)).

−→

Figure 3.5: Example of type D6

(iii). By (i) and (ii), the algebras BrD(Bn) and SBr(Dn+1) are isomorphic Z[δ±1]-algebras.The latter is free (as stated above) and the definition of the former shows that its rankis |T |n+1| + |T

|=n+1| + |T

=n ∩ T 0

n+1|. A simple counting argument gives |T |n+1| = 2n · n!!,

|T |=n+1|= 2n · n!!− n!, and |T=n ∩ T 0n+1|= (n+1)!!− (n+1)!. We conclude that the rank of

BrD(Bn) is equal to f (n).

3.3 ADMISSIBLE ROOT SETS AND THE MONOID ACTION 59

For n = 3, the eight Brauer diagrams in ψφ(F) of BrD(D4) ([13, Section 4]) aredepicted in Figure 3.6.

1 e2e0 e1e0e1

e0e1 e1e0 e0e2 e2r1e0e1e2

θδ−1 θδ−1

θδ−1 θδ−1 θδ−1 θδ−1

Figure 3.6: The images of F under ψφ

3.3 Admissible root sets and the monoid action

In this section, we recall some facts about the root system associated to the Brauer algebraof type Dn+1, introduce the definition of admissible root sets of type Bn and describe someof their basic properties. Also by use of Chapter 2, we give a monoid action of BrM(Bn)on the admissible root sets.

Definition 3.3.1. By Φ and Φ+, we denote the root system of type Dn+1 and its positiveroots, and where Φ+ can be realized by vectors ε j ± εi with j > i in Rn+1 with {εi}n+1

i=1being the canonical orthonormal basis. The simple roots are α1 = ε1 + ε2 and αi =εi−εi−1 for i = 2, . . . , n+1. If α is a root of Φ+, then α∗ denotes its orthogonal mate, thatis, α∗ = ε j+εi if α= ε j−εi ∈ Φ+, and (α∗)∗ = α. When n≥ 4, this is the unique positiveroot orthogonal to α and all other positive roots orthogonal to α (see [12, Definition2.6]). The diagram automorphism σ on W (Dn+1) is induced by a linear isomorphism σon Rn+1, where σ is the orthogonal reflection with root ε1. We define p : Rn+1 → Rn+1

by p(x) = (σ(x)+ x)/2. The set Ψ= p(Φ) forms a root system of the Weyl group of typeBn with β0 = ε2 = (α1 +α2)/2 and βi = εi+2 − εi+1 = αi+2 for i = 1, . . . , n− 1 as simpleroots, and Ψ+ = p(Φ+) consists of all positive roots of the Weyl group of W (Bn). A rootβ ∈ Ψ is called a long root if |β | =

p2, and called a short root if |β | = 1. The subset of

Ψ+ consisting of all short roots is {εi}n+1i=2 .

Recall the definition of admissible root sets in Definition 1.6.4.

Remark 3.3.2. The mutually orthogonal root set B1 = {ε2,ε3} is not admissible, forp−1(B1) ∩ Φ = {α1, α2, α1 + α3, α2 + α3} is not admissible. Although B2 = {α1,α4} isadmissible in Φ+, the set p(B2) is not admissible, as p−1p(B2) = {α1,α2,α4} is not admis-sible.

60 TYPE Bn

By use of the description of admissible sets of type D in [12, Section 4], the admissiblesets of type B can be classified as indicated below.

Proposition 3.3.3. The admissible root subsets of Ψ+ can be divided into the followingthree types.

(1) The set consists of long roots and none of their orthogonal mates are in this set. Itbelongs to the W (Bn)-orbit of Zt = {βn−1−2i | 0 ≤ i < t}, for some t with 0 ≤ t <(n+ 1)/2.

(2) The set consists of long roots and their orthogonal mates. It belongs to the W (Bn)-orbit of Zt = {βn−1−2i ,β

∗n−1−2i | 0≤ i < t}, for some t with 1≤ t < (n+ 1)/2.

(3) The set has only one short root, and for each long root that it also contains, it alsocontains its orthogonal mate. It belongs to the W (Bn)-orbit of Zt = {βn−1−2i ,β

∗n−1−2i |

0≤ i < t − 1} ∪ {β0}, for some t with 1≤ t ≤ (n+ 1)/2.

Lemma 3.3.4. The cardinalities of the W (Bn)-orbits of Zt , Zt , and Zt in the above are2t� n

2t

�

t!!, n� n−1

2t−2

�

(t − 1)!!, and� n

2t

�

t!!, respectively.

Proof. The orbits of Zt , Zt , and Zt correspond to, respectively,

(1) the diagrams with exactly t decorated horizontal strands at the top in BrD(Dn+1)none of which has end 1,

(2) the diagrams with exactly t horizontal strands at the top without decoration one ofwhich has end point 1,

(3) the diagrams with exactly t horizontal strands at the top without decoration andnone of which has end point 1.

Therefore, the sizes are easily seen to be as stated.

Recall formula (1.6.30). We write e∗β = eβ∗ and e∗i = eβ∗i for β ∈Ψ+ and 1≤ i ≤ n−1in accordance with the definition before Lemma 3.2.4.

Lemma 3.3.5. If {γ1, γ2} is a subset of some admissible root set, then

eγ1eγ2= eγ2

eγ1.

Proof. In view of Proposition 3.3.3, the proof can be reduced to a check in the followingthree cases

(γ1,γ2) = (β0,β2), (β1,β3), or (β1,β∗1).In the first two cases the lemma holds due to (1.6.6). In the third case it holds by (1.6.12)and (3.2.5).

Let X ⊂Ψ+ be a subset of some admissible root set. Then we define

eX = Πβ∈X eβ . (3.3.1)

In view of Lemma 3.3.5, it is well defined.

3.3 ADMISSIBLE ROOT SETS AND THE MONOID ACTION 61

Lemma 3.3.6. Let X ⊂Ψ+ be a mutually orthogonal root set and X exists. Then

eX = δ|X\X |eX .

Proof. If X is in the W (Bn)-orbit of Zt , then it is trivial. If X is in the W (Bn)-orbit of Zt orZt , then X can be transformed into a subset of Zt or Zt by the action of some element ofW (Bn). Hence we just consider subsets of Zt and Zt . Now

e0e2e∗2 = e0e2r1r0(r1e2r1)r0r1(1.6.9)= e0e2r1(r0r2)e1r2r0r1

(1.6.4)= e0(e2r1r2)r0e1(r2r0)r1

(1.5.1)+(1.6.4)= (e0e2)(e1r0e1r0)r2r1

(1.6.6)+(3.2.5)= e2(e0e1e0)e1r2r1

(1.6.15)= δ(e2e0)e1r2r1

(1.6.6)= δe0(e2e1r2r1)

(1.5.4)+(1.6.1)= δe0e2,

e1e∗1e3e∗3(3.2.5)= e1e0(e1e3)r2r1r0r1r2e3r2r1r0r1r2

(1.6.6)= e1e0e3(e1r2r1)r0r1r2e3r2r1r0r1r2

(1.5.1)= e1e0e3e1(e2r0)r1r2e3r2r1r0r1r2

(1.6.5)= e1e0(e3e1r0)(e2r1r2)e3r2r1r0r1r2

(1.6.5)+(1.6.6)+(1.5.1)= e1e0e1r0e3e2(e1e3)r2r1r0r1r2

(1.6.17)= e1e0e1e3e2e3(e1r2r1)r0r1r2

(1.5.1)= e1e0e1(e3e2e3)e1e2r0r1r2

(1.5.5)= e1e0e1(e3e1)e2r0r1r2

(1.6.6)= e1e0(e1e1)e3e2r0r1r2

(1.6.3)= δe1e0e1(e3e2r0)r1r2

(1.6.5)= δ(e1e0e1r0)e3e2r1r2

(1.6.17)= δe1e0(e1e3)e2r1r2

(1.6.6)= δe1e0e3(e1e2r1r2)

(1.5.4)= δe1e0e3e1(r2r2)

(1.6.1)= δe1e0e3e1

(3.2.5)= δe1e∗1e3.

This proves the lemma for |X | = 2, 3. By applying induction on |X |, we see the lemmaholds.

Remark 3.3.7. Recall the monoid action defined in Definition 1.3.2. In [13, Section 4],a root εi − ε j (εi + ε j) with 1 ≤ j < i ≤ n+ 1 is represented as a (decorated) horizontalstrand from i to j at the top of the diagram. This way, the above monoid action canbe given a diagram explanation in which admissible sets are represented by the set ofhorizontal strands at the top of a diagram.

As in Proposition 2.4.5, there is a monoid action of BrM(Bn) onAσ under the compo-sition of the above action of Br(Dn+1) and φ. It gives an action of Br(Bn) on the collectionof admissible subsets of Ψ+. This action also has a diagrammatic interpretation obtainedfrom viewing the admissible subsets of Ψ+ as tops of symmetric diagrams by use of p−1.For example,

e0{β1} = p(φ(e0)p−1{β1}) = p(E1E2{α3})

= p({α1,α2}) =�

α1 +α2

2

�

= {β0}.

62 TYPE Bn

3.4 An upper bound on the rank

In Theorem 3.4.8 of this section, normal forms for elements of the Brauer monoid BrM(Bn)will be given, and in Corollary 3.4.18, a spanning set for Br(Bn) of size f (n) will be given.The rank of Br(Bn) will be proved to be f (n) as a consequence. These results will providea proof of Theorem 3.1.2.

The normal forms of monomials in Br(Bn) will be parameterized by a set F of ele-ments f (i)t to be defined below, a general form being a = uf (i)t v for certain elementsu, v ∈W (Bn). In fact, F is a set of representatives for the W (Bn)-orbits of the admissiblesets a(;) and aop(;) which appear in the guise of horizontal strands at top and bottom,respectively, as discussed in Remark 3.3.7. Recall g = e2r1e0e1e2 from (3.2.8).

Definition 3.4.1. We define

f (1)t := eZt=

t∏

i=1

en+1−2i , 0≤ t < (n+ 1)/2,

f (2)t := eZt=

t∏

i=1

en+1−2ie∗n+1−2i , 1≤ t ≤ n/2,

f (3)t := eZt, 1≤ t ≤ (n+ 1)/2,

f (4)t := g f (2)t−1, 2≤ t ≤ (n− 1)/2,

f (5)t := e0e1 f (2)t−1, 2≤ t ≤ n/2,

f (6)t := e1e0 f (2)t−1, 2≤ t ≤ n/2,

and f (4)1 := g, f (5)1 := e0e1, f (6)1 := e1e0. Furthermore, we denote by F the set of allelements f (i)t , and write M = δZW (Bn)FW (Bn).

The following statement is immediate from the definition of M .

Lemma 3.4.2. The set M is closed under multiplication by any element from W (Bn).

Note that f (3)t = e0 f (2)t−1, for 2≤ t ≤ [(n+1)/2]. The set F is closed under the naturalanti-involution x 7→ xop of Proposition 1.6.3.

For n= 3, we find f (1)0 = 1, f (1)1 = e2, f (3)1 = e0, f (6)1 = e1e0, f (5)1 = e0e1, f (2)1 = e2e∗2 =r1r2(e1e0e1)r2r1, f (3)2 = δe0e2, and f (4)1 = g = e2r1e0e1e2, which fits (up to conjugationby a Coxeter element) with the eight elements of F depicted in Figure 3.6.

Part of Theorem 3.4.8 states that each member of M has a unique decomposition, theother part states that M coincides with BrM(Bn). The first part, whose proof takes all ofthis section up to the statement of the theorem, is devoted to giving a normal form foreach element of M , and involves a narrowing down of the possibilities for the elementsof W (Bn) in a normal form at both sides of f (i)t . The second part is carried out after thestatements of Theorem 3.4.8, where it is shown that M is invariant under multiplicationby ri and ei for each i ∈ {0, . . . , n− 1}. As F = Fop, it suffices to consider multiplicationfrom the left. As eiuf (i)t = ueβ f (i)t , where β = u−1βi , it suffices to verify that eγ f (i)tbelongs to M for each γ ∈Ψ+. This is the content of Lemmas 3.4.9–3.4.16.

3.4 AN UPPER BOUND ON THE RANK 63

Definition 3.4.3. Let X be an admissible subset of Ψ+. Recall the action of W (Bn) on alladmissible subsets of Ψ+ as explained in Remark 3.3.7. The stabilizer of X in W (Bn) isdenoted by N(X ). We select a family DX of left coset representatives of N(X ) in W (Bn),so |DX | is the size of W (Bn)-orbits of X . We simplify DZt

, DZt, DZt

, NZt,NZt

, NZtto D(1)t ,

D(2)t , D(3)t , N (1)t N (2)t , N (3)t respectively.In order to identify the part of W (Bn) that commutes with f (i)t , we introduce the

following subgroups of W (Bn).

C (1)0 = W (Bn),

C (1)t =¬

r∗n−1, r0, r1, . . . , rn−1−2t

¶

, 1≤ t ≤ [n/2],

C (2)t =

r1, r2, . . . , rn−1−2t�

, 1≤ t ≤ [n/2],

C (3)t =

r2, r3, . . . , rn+1−2t�

, 1≤ t ≤ [(n+ 1)/2],

C (4)t =

r4, r5, . . . , rn+1−2t�

, 1≤ t ≤ [(n− 1)/2],

C (5)t = C (6)t =

r3, r4, . . . , rn+1−2t�

, 1≤ t ≤ [n/2].

For 0≤ t ≤ [n/2], write

A(1)t =

rn−2i rn+1−2i rn−1−2i rn−2i�t−1

i=1 n

rn+1−2i�t

i=1 ,

W (1)t =

r0, r1, . . . , rn−1−2t�

×¬

r∗n+1−2i

¶t

i=1.


A(2)t =D

rεn+2−2t, A(1)t

E

n¬

r∗n+1−2i

¶t

i=1,

W (2)t =

r0, r1, . . . , rn−1−2t�

.

For 1≤ t ≤ [(n+ 1)/2], write

A(3)t =¬

rεn+4−2t, rn−2i rn+1−2i rn−1−2i rn−2i

¶t−2

i=1n¬

rn+1−2i , r∗n+1−2i

¶t−1

i=1×

r0�

,

W (3)t =

r1r0r1, r2, r3, . . . , rn+1−2t�

.

For 1≤ t ≤ [(n− 1)/2], write

A(4)t =D

r1r0r1, rεn+4−2t−ε4r2rn+3−2t rεn+4−2t−ε4

, A(1)t−1, r2, r∗2 , {rn+1−2i , r∗n+1−2i}t−1i=1 , r0

E

,

W (4)t =

D

rε5, C (4)t

E

.


A(5)t = A(3)t ×

r1r0r1�

,

W (5)t =

r2r1r0r1r2, r3, r4, . . . , rn+1−2t�

.

We first determine the structure of these subgroups.

64 TYPE Bn

Lemma 3.4.4. (i) For 1≤ t ≤ [n/2], we have

C (1)t∼= W (Bn−2t)×W (A1),

A(1)t∼= W (At−1)n (W (A1))

t ,

W (1)t

∼= W (Bn−2t)(W (A1))t .

(ii) For 1≤ t ≤ [n/2], we have

C (2)t∼= W (An−1−2t),

A(2)t∼= W (Bt)n (W (A1))

2t ,

W (2)t

∼= W (Bn−2t).

(iii) For 1≤ t ≤ [(n+ 1)/2], we have

C (3)t∼= W (An−2t),

A(3)t∼= W (Bt−1)n (W (A1))

2t−1,

W (3)t

∼= W (Bn+1−2t).

(iv) For 1≤ t ≤ (n− 1)/2, we have

A(4)t∼= W (Bt)nW (A1)

2t+1,

W (4)t

∼= W (Bn−1−2t),

A(4)t ∩W (4)t = {1},

C (4)t∼= W (An−2−2t).

Furthermore, the group A(4)t normalizes W (4)t .

(v) For 1≤ t ≤ n/2, we have

A(5)t∼= W (Bt−1)n (W (A1))

2t ,

W (5)t

∼= W (Bn−2t),

A(5)t ∩W (5)t = {1},

C (5)t∼= W (An−1−2t).

Furthermore, A(5)t normalizes W (5)t .

Proof. With the diagram representation of Figure 3.3, the argument is analogous toLemma 2.5.4.

Lemma 3.4.5. For i = 1, 2, 3, the subgroup N (i)t in W (Bn) is a semiproduct of W (i)t and

A(i)t .

Proof. The semidirect group structure of these subgroups can be proved by use of thediagram representation of Figure 3.3. The normalizer (stabilizer) claims follow fromLagrange’s Theorem and Lemma 3.3.4.


Definition 3.4.6. We need a few more subgroups and coset representatives. For i = 4,5, let N (i)t = A(i)t W (i)

t , and D(i)t be its left coset representatives in W (Bn). The sets Zt andZt−1∪{β1,β∗1} are conjugate by τ= rεn+2−2t−ε3

r1rn+1−2t rεn+2−2t−ε3, and τC (2)t τ

−1 = C (6)t . At

the same time we have τ f (2)t τ−1 = δe1 f (2)t−1. Let N (6)t be the stabilizer of Zt−1∪{β1,β∗1} in

W (Bn) and D(6)t be a set of left coset representatives of N (6)t in W (Bn). Let A(6)t = τA(2)t τ−1

and W (6)t = τW (2)

t τ−1.Finally, for i = 1, 2, 3, 4, let D(i)t,L = D(i)t,R = D(i)t , D(5)t,L = D(6)t,R = D(5)t , and D(6)t,L = D(5)t,R =

D(6)t .

Proposition 3.4.7. In Br(Bn), the following properties hold for all i ∈ {1, . . . , 6}.

(i) For each x ∈ N (i)t we have x f (i)t = f (i)t x.

(ii) For each a ∈ A(i)t , we have a f (i)t = f (i)t .

(iii) For each b ∈W (i)t there exists some c ∈ C (i)t , such that b f (i)t = c f (i)t .

(iv) For each c ∈ C (i)t we have c f (i)t = f (i)t c.

As a result, each monomial in M can be written in the normal form

uf (i)t vw,

for some u ∈ D(i)t,L , w ∈ (D(i)t,R)op, v ∈ C (i)t , and i ∈ {1, . . . , 6}.

Proof. (i). For i ∈ {1,2, 3}, this follows from Definition 3.3.1 and Lemma 3.4.5. Fori ∈ {4,5, 6}, observe that N (i)t is the (semi-direct) product of W (i)

t and A(i)t , so (i) willfollow from (ii), (iii), and (iv) if we prove (ii), (iii), and (iv).

(ii). We use the following three equalities.

ri ri−1ri+1riei−1ei+1 = ei−1ei+1, for i> 1 (3.4.1)

r0eie∗i = eie

∗i , (3.4.2)

riei = ei . (3.4.3)

The first holds as

ri ri−1ri+1ri(ei−1ei+1)(1.6.6)= ri ri−1(ri+1riei+1)ei−1

(1.6.8)= (ri ri−1ei)(ei+1ei−1)(1.6.8)= (ei−1eiei−1)ei+1

(1.5.5)= ei−1ei+1,

the second equality is from Lemma 3.2.4, and the third equality follows from definition.Now for the proof that the lemma holds for each generator, the formulas (3.4.1)–(3.4.3)or their conjugations can cover all possible cases.

(iii). The difference of generators of W (1)t and C (1)t is made up of {r∗n+1−2i}

ti=2, which are

conjugate to r∗n−1 by some elements in A(1)t , for example

rn−2rn−3rn−1rn−2β∗n−3 = β

∗n−1.

66 TYPE Bn

Therefore (iii) for f (1)t follows from (ii). We can derive it for f (2)t and f (3)t by applyingLemma 3.2.4.

For i = 4, recall that g = e2r1e0e1e2. By use of Lemma 3.2.5 and

rεn+4−2t−ε4r2rn+3−2t rεn+4−2t−ε4

e2en+3−2t(3.4.1)= e2en+3−2t ,

(ii) and (iii) for f (4)t can be obtained.For i = 5, the (ii) holds in view of r1r0r1e0 = e0 and the argument of (ii) for f (3)t ;

the(iii) holds because of

r2r1r0r1r2e0(1.6.5)= r2(r1r0r1e0)r2

(1.6.11)= r2e0r2

(1.6.5)= e0

and (iii) for f (3)t .When i = 6, (ii), (iii) hold naturally for (ii), (iii) of f (2)t .As for the final statement, Note that, if w ∈ W , we can write w = vn with v ∈ D(i)t

and n ∈ N (i)t ; but n = ba with b ∈W (i)t and a ∈ A(i)t ; so, by (iii), w f (i)t = vb f (i)t = v f (i)t c

for some c ∈ C (i)t . Using the opposition involution of Proposition 1.6.3, we can finish

for i = 1, 2,3, 4 as�

f (i)t

�op= f (i)t . The other two cases can be treated similarly, so we

restrict ourselves to i = 5.Applying the results obtained so far to the image under opposition of f (5)t z for z ∈W ,

we find u ∈ D(6)t = D(5)t,R and d ∈ C (6)t such that ( f (5)t z)op = z−1 f (6)t = u−1 f (6)t d−1, so

f (5)t z = d f (5)t u. As d ∈ C (6)t = C (5)t (see Definition 3.4.3), this expression blends well withv f (5)t c for w f (5)t to give the required normal form for w f (5)t z.

(iv). This follows from a straightforward check for each the generators of C (i)t .

We now come to the complete normal forms result by replacing M in the last state-ment of Proposition 3.4.7 with BrM(Bn).

Theorem 3.4.8. Up to powers of δ, each monomial in BrM(Bn) can be written in the normalform

uf (i)t vw,

for some u ∈ D(i)t,L , w ∈ (D(i)t,R)op, v ∈ C (i)t , and f (i)t ∈ F.

In view of Proposition 3.4.7 and Lemma 3.4.2, for the proof of the theorem it remainsto consider the products of the form eβ f (i)t . This is done in Lemmas 3.4.9—3.4.16.

Lemma 3.4.9. For f (1)t and f (2)t , the following statements hold.

(i) If i < n− 2t, then there exist r, s ∈W (Bn) such that

eεi+2f (1)t = δt r f (3)t+1r−1 and eεi+2

f (2)t = s f (3)t+1s−1.

(ii) If i ≥ n− 2t, then there exists s ∈W (Bn) such that

eεi+2f (1)t = δt−1s f (5)t s−1 and eεi+2

f (2)t = s f (5)t s−1.


As a consequence, for each short root β of Ψ+ and each i ∈ {1, 2}, the monomial eβ f (i)tbelongs to M.

Proof. (i). We have eεi+2f (1)t = eB = δ−t eB, where B = {εi+2} ∪ Zt and B = {εi+2} ∪ Zt

is on the W (Bn)-orbit of Zt+1, hence the first equality. The second equality holds by asimilar argument.

(ii). First consider the case i = n− 2t. For s = rεi+3−ε3rεi+2−ε2

, we have

sεi+2 = β0, sβn+1−2t = β1, sβ∗n+1−2t = β∗1 ,

and so

s(Zt \ {βn+1−2t}) = (Zt \ {βn+1−2t}),s(Zt \ {βn+1−2t ,β

∗n+1−2t}) = (Zt \ {βn+1−2t ,β

∗n+1−2t}).

Therefore

seεi+2f (1)t s−1 = seεi+2

s−1seβn+1−2ts−1s f (1)t−1s−1 = e0e1 f (1)t−1 = δ

t−1 f (5)t ,

and similarly for eεi+2f (2)t instead of eεi+2

f (1)t .Next, consider i > n − 2t. Either βi or βi+1 ∈ Zt , suppose first βi+1 ∈ Zt . Now

βi+1 ∈ Zt ⊂ Zt , and ri ri−1ri+1ri interchanges βi+1 and βi−1 as well as β∗i+1 and β∗i−1;moreover, it fixes all other elements of Zt and Zt . Hence ri ri−1ri+1riεi+2 = εi and thelemma holds by induction on i. For βi ∈ Zt ⊂ Zt , note that ri{εi+2} = {εi+1} and thatri keeps Zt and Zt invariant. It follows that, by conjugation with ri ri−1ri+1ri and ri , thetwo equalities are brought back to the cases for i− 2 and i− 1, respectively.

As for the final statement, note that each short root is of the form ε j for some j ∈{2, . . . , n+ 1}.

Let W =W (Bn).

Lemma 3.4.10. If β is a long root in Ψ+, then

eβ f (1)t ∈ δZW f (1)t ∪δZW f (1)t+1W ∪δZ f (2)t ,

eβ f (2)t ∈ δZW f (2)t ∪δZW f (2)t+1W.

Proof. Let’s consider eβ f (1)t . First, if β ∈ Zt , then eβ f (1)t = δ f (1)t . Second, if β is anorthogonal mate of any element in Zt , we have

eβ f (1)t = e{β}∪Zt= δt−1e{β}∪Zt

= δt−1 f (2)t ,

Third, if β and β ′ ∈ Zt ∪ Zt are two long roots and not orthogonal to each other. Then

eβ eβ ′(1.6.8)= rβ ′ rβ eβ ′ , therefore eβ f (1)t ∈ δZW f (1)t . Fourth, β is not in the above three cases,

hence {β} ∪ Zt will be on the W -orbit of Zt+1, therefore we have eβ f (1)t ∈ δZW f (1)t+1W .

The second claim of eβ f (2)t holds by a similar argument.

68 TYPE Bn

Lemma 3.4.11. For each long root β ∈Ψ+, we have eβ f (3)t ∈ M.

Proof. If β is not orthogonal to all roots of Zt−1, then eβ f (3)t = eβ f (2)t−1e0 by applyingLemma 3.4.10 and Lemma 3.4.2.

If β is orthogonal to Zt−1 and β0, there exists a r ∈ N (3)t , which does not move anyelement in Zt−1 and β0 and rβ = βn+1−2t ; hence eβ f (3)t = r f (3)t+1r−1.

If β is orthogonal to Zt−1 and not orthogonal to β0, then we can find an elementr ∈ N (3)t such that re1r−1 = eβ , which implies that eβ f (3)t = r−1e1e0 f (2)t−1r ∈ W f (6)t W ,therefore our lemma holds.

Lemma 3.4.12. For each short root β ∈Ψ+, we have eβ f (3)t , eβ f (5)t ∈ M.

Proof. There is an index i such that β = ri ri−1 · · · r1β0. Now

eβ e0 = ri ri−1 · · · r1e0r1(r2 · · · rie0)(1.6.5)= ri ri−1 · · · r1(e0r1e0)r2 · · · ri

(1.6.14)= δri ri−1 · · · r1e0r2 · · · ri

(1.6.5)= δri ri−1 · · · r1r2 · · · rie0

∈ δWe0,

and the lemma follows as both f (3)t and f (5)t begin with e0.

Lemma 3.4.13. Let β ∈Ψ+ be a long root.

(i) If β is not orthogonal to Zt−1, then eβ f (5)t ∈ δZW f (5)t .

(ii) If β = ε j ± εi , with 2< i < j < n+ 4− 2t, then eβ f (5)t ∈ δZW f (5)t+1W.

(iii) If β = ε j ± ε3, with 3< j < n+ 4− 2t, then eβ f (5)t ∈ δZW f (3)t+1W.

(iv) If β = ε j ± ε2, with 2< j < n+ 4− 2t, then eβ f (4)t ∈ δZW f (5)t+1W.

(v) If β = β1, then eβ f (5)t = e1 f (5)t = r f (2)t r−1, for some r ∈W.

Thus for each β ∈Ψ+ the monomial eβ f (5)t belongs to M.

Proof. As rεi(ε j + εi) = ε j − εi and {rεi

}n+1i=2 ⊂ N (5)t , we only need consider β = ε j − εi .

Case (i) can be checked easily. Case (v) holds as

e1 f (5)t = e1e0e1 f (2)t−1(1.6.12)= e1e∗1 f (2)t−1 = e{β1,β∗1}∪Zt−1

,

where {β1,β∗1} ∪ Zt−1 are on the W -orbit of Zt .

After conjugation by suitable elements of N (5)t , we can restrict ourselves to β =βn+1−2t , β2, β1+β2 for cases (ii), (iii), and (iv), respectively. Case (ii) can be proved eas-ily. Case (iii) holds as e2 f (5)t = e0e2e1 f (2)t−1 = e0e2r1r2 f (2)t−1 = e0e2 f (2)t−1r1r2, and e0e2 f (2)t−1

is conjugate to e0 f (2)t by some element of W (Bn). Case (iv) holds as eβ1+β2e0e1 =

r1e2r1e0e1(1.5.4)= r1e2r1e0e1e2r1r2 = r1 gr1r2.


Lemma 3.4.14. For each short root β ∈Ψ+, we have eβ f (6)t ∈ M.

Proof. If β is equal to β0 or ε3, the lemma holds as

e0 f (6)t = e0e1e0 f (2)t−1(1.6.15)= δe0 f (2)t−1 = δ f (3)t ,

eε3f (6)t = r1e0r1e1e0 f (2)t−1 = δr1e0 f (2)t−1 = δr1 f (3)t .

If β = εi+2 for 1< i < n, then eβ = ri · · · r2r1e0r1r2 · · · ri . Hence

eβ f (6)t = ri · · · r2r1e0r1r2 · · · ri f (6)t

(1.6.5)= ri · · · r2r1(e0r1r2e1e0)r3 · · · ri f (2)t−1

(1.6.8)= ri · · · r2r1(e0e2e1e0)r3 · · · ri f (2)t−1

(1.6.6)+(1.6.15)= δri · · · r2r1e0e2r3 · · · ri f (2)t−1

= δri · · · r2r1e0r3 · · · ri(r3 · · · ri)−1e2r3 · · · ri f (2)t−1

(1.6.5)= δri · · · r2r1r3 · · · rie0(r3 · · · ri)

−1e2r3 · · · ri f (2)t−1,

∈ We0eεi+2−ε3f (2)t−1,

⊆ We0W ( f (2)t ∪ f (2)t−1)W by Lemma 3.4.10

⊆ W ( f (3)t ∪ f (3)t+1 ∪ f (5)t ∪ f (5)t−1)W by Lemma 3.4.9

⊆ M .

Lemma 3.4.15. If β ∈Ψ+ is a long root, then eβ f (6)t belongs to M.

Proof. If β = ε j+2 ± εi+2 with 1< i < j < n, then the lemma holds as

eβ e1e0 f (2)t−1 = e1(e0(eβ f (2)t−1))

∈ e1e0(δZW f (2)t−1W ∪δZW f (2)t W ) by Lemma 3.4.10

⊆ δZ(e1W f (3)t W∪W f (5)t−1W∪W f (3)t+1W∪W f (5)t W ) by Lemma 3.4.9

⊂ M by Lemma 3.4.11 and Lemma 3.4.13.

Otherwise, β is not orthogonal to β1, and so eβ e1e0 f (2)t−1(1.6.8)= r1rβ e1e0 f (2)t−1 if β 6∈ {β1, β∗1},

or δe1e0 f (2)t−1 if β ∈ {β1, β∗1}. Therefore, the lemma holds.

Finally, we deal with f (i)t for i = 4.

Lemma 3.4.16. For each β ∈Ψ+, the monomial eβ f (4)t belongs to M.

70 TYPE Bn

Proof. First assume that β is a short root. If β = εi+2 with i > 2, then

eβ f (4)t = eβ g f (2)t−1 = ri · · · r4r3r2r1e0r1r2r3 · · · ri g f (2)t−1

(1.6.5)= ri · · · r4r3r2r1(e0r1r2r3 g)r4 · · · ri f (2)t−1

(3.2.14)= δri · · · r4r3r2r1(e0(e1(e3r2 · · · ri f (2)t−1)))

∈ δZWe0 f (2)t ′ W by Lemma 3.4.10

⊆ M by Lemma 3.4.9

If β = ε4, ε3, or β0, the same argument as above can be applied with (3.2.15) and(3.2.12) in Lemma 3.2.5.

Next assume β is a long root. If β = ε j+2 ± εi+2 with 2 < i < j < n, then eβ g f (2)t−1 =geβ f (2)t−1. If β is not orthogonal to Zt−1 (see Proposition 3.3.3), then either eβ f (2)t−1 =rs rβ f (2)t−1, for some βs ∈ Zt−1 not orthogonal to β , or eβ f (2)t−1 = δ f (2)t−1. Now the lemmaholds as rs rβ g = grs rβ . Otherwise, j < n + 2 − 2t, and we can find some element

r ∈ N (2)t−1 such that rβ = βn+1−2t . Then eβ f (2)t−1 = r−1en+1−2t r f (2)t−1 = r−1en+1−2t f (2)t−1r =δ−1r−1 f (2)t r, for r−1 commutes with g, hence the lemma holds by Lemma 3.4.2.

If β = ε j+2 ± εi+2 with 0 < i ≤ 2 ≤ j < n, the root β is not orthogonal to β2 orβ ∈ {β2, β∗2}, hence eβ g = r2rβ g or δg, which implies that the lemma holds by Lemma3.4.2.

If β = ε j+2± ε2, then r0β = β∗ and r0 ∈ N (2)t , so it suffices to consider β = ε j+2− ε2.If 2< j, then

eβ f (2)t = eβ g f (2)t−1 = r j · · · r4r3r2e1r2r3 · · · r j g f (2)t−1

= r j · · · r4r3r2(e1r2r3 g)r4 · · · r j f (2)t−1

(3.2.16)= r j · · · r4r3r2(e1(e0r1r2r3(e2r4 · · · r j f (2)t−1)))

∈ δZWe1e0W f (2)t W by Lemma 3.4.10

⊆ δZWe1W f (3)t ′ W ∪δZWe1W f (5)t ′ W by Lemma 3.4.9

⊆ M , by Lemma 3.4.11 and Lemma 3.4.13

and we are done. The remaining two cases are β = β1 and β = β1 + β2. These followreadily from e1e2 = r2r1e2 and r2e1r2e2 = r1e2. This proves the lemma.

By means of Lemmas 3.4.9—-3.4.16, we have shown that eβ f (i)t ∈ M for each β ∈Ψ+and each i ∈ {1, . . . , 6}, which suffices to complete the proof of Theorem 3.4.8.

We proceed to give an upper bound for the rank of the Brauer algebra Br(Bn). ByTheorem 3.4.8, the upper bound is given by

6∑

i=1

|D(i)t,L ||D(i)t,R||C

(i)t |. (3.4.4)

Table 3.1 lists the cardinalities of D(i)t,L , D(i)t,R, and C (i)t .


Table 3.1: Cardinalities of coset and centralizers

D(1)0 1 C (1)0 2nn!D(1)t 2t� n

2t

�

t!! C (1)t 2n+1−2t(n− 2t)!

D(2)t� n

2t

�

t!! C (2)t (n− 2t)!

D(3)t n� n−1

2t−2

�

(t − 1)!! C (3)t (n+ 1− 2t)!

D(4)t n�n−1

2t

�

t!! C (4)t (n− 1− 2t)!

D(5)t n(n− 1)� n−2

2t−2

�

(t − 1)!! C (5)t (n− 2t)!

D(6)t� n

2t

�

t!! C (6)t (n− 2t)!

Lemma 3.4.17. For 0< t ≤ (n+ 1)/2,

6∑

i=2

|D(i)t,L ||D(i)t,R||C

(i)t |=

��

n+ 1

2t

�

t!!�2

(n+ 1− 2t)!.

Proof. Recall our M (i), i = 2, . . . , 6, in the proof of Theorem 3.2.8, and let M (i)t be thesubset of diagrams M (i) with t horizontal strands at the top of their diagrams. TheseM (i)t consist of all possible diagrams with t horizontal strands in T=n+1 ∩ T 0

n+1. The countof classical Brauer diagrams (of [4]) related to the Brauer monoid of type Br(An) witht horizontal strands at the top can be conducted as follows. First choose 2t points atthe top (bottom) and make t horizontal strands; the remaining n+ 1− t vertical strandscorrespond to the elements of the Coxeter group of type W (An−t). Therefore the righthand side of the equality is the number of all possible diagrams in T=n+1 ∩ T 0

n+1 with thorizontal strands at the top, and so equals

6∑

i=2

|M (i)t |=��

n+ 1

2t

�

t!!�2

(n+ 1− 2t)!.

We compute |M (i)t | for i = 2, . . . , 6.For i = 2, we just count as above with n+ 1 replaced by n, so

|M (2)t |=��

n

2t

�

t!!�2

(n− 2t)!= |D(2)t |2|C (2)t |= |D

(2)t,L ||D

(2)t,R||C

(2)t |.

For i = 5, we first choose two points from the top n+1 points except 1 for the horizontalstrand from 1 and the vertical strand from 1, and we choose 2(t − 1) points at the topfrom the remaining n+ 1− 3 points at the top for t − 1 horizontal strands and 2t pointsat the bottom n+1 points except 1 for t horizontal strands, and then the vertical strandsbetween the remaining n − 2t points at the top and the remaining n − 2t points atthe bottom will be corresponding to elements of the Coxeter group of type W (An−2t−1).Therefore,

|M (5)t |= n(n− 1)�

n− 2

2t − 2

�

(t − 1)!!�

n

2t

�

t!!(n− 2t)!

72 TYPE Bn

= |D(5)t ||D(6)t ||C

(5)t |= |D

(5)t,L ||D

(5)t,R||C

(5)t |.

By reversing the top and bottom, we obtain a one to one correspondence between M (5)t

and M (6)t ; it follows that

|M (6)t |= |M(5)t |= |D

(6)t,L ||D

(6)t,R||C

(6)t |.

For i = 4, we first choose one point from the bottom (top) n + 1 points except 1 (1)for the vertical strand from 1 (1); the remaining count of horizontal strands and othervertical stands is as in the classical case after replacing n+ 1 by n− 1; hence

|M (4)t |= n2��

n− 1

2t

�

t!!�2

(n− 1− 2t)!= |D(4)t |2|C (2)t |= |D

(4)t,L ||D

(4)t,R||C

(4)t |.

For i = 3, we first choose one point at the top (bottom) distinct from 1 (1) for thehorizontal strand from 1 (1); the remaining count of other horizontal strands and verticalstands is as in the classical case after replacing n+ 1 by n− 1 and t by t − 1; it followsthat

|M (3)t | = n2��

n− 1

2t − 2

�

(t − 1)!!�2

(n+ 1− 2t)!= |D(3)t |2|C (3)t |

= |D(3)t,L ||D(3)t,R||C

(3)t |.

The equality of the lemma now follows from the above 5 equalities for M (i)t .

Corollary 3.4.18. The algebra Br(Bn) has a spanning set over Z[δ±1] of size at most f (n).

Proof. By (3.4.4), the rank of Br(Bn) is at most

|W (Bn)|+[ n

2]

∑

t=1

|D(1)t,L ||D(1)t,R||C

(1)t |+

[ n+12]

∑

t=1

��

n+ 1

2t

�

t!!�2

(n+ 1− 2t)!

= 2n · n!+ 2n

[ n2]

∑

t=1

��

n

2t

�

t!!�2

(n− 2t)!+[ n+1

2]

∑

t=1

��

n+ 1

2t

�

t!!�2

(n+ 1− 2t)!.

From [35], it follows that

[ k2]

∑

t=0

��

k

2t

�

t!!�2

(k− 2t)!= k!!.

By applying this for k = n, n+1 to the last two summands of the above equality, we obtainthat the rank of Br(Bn) is at most 2n+1 · n!!− 2n · n!+ (n+ 1)!!− (n+ 1)!= f (n).

We end this section with a proof of Theorem 3.1.2. By Corollary 3.4.18 there is aspanning set of Br(Bn) of size f (n). By Theorem 3.2.8, this set maps onto a spanningset of SBr(Dn+1) of size at most f (n). Moreover, by the same theorem, SBr(Dn+1) is afree algebra over Z[δ±1] of rank f (n). This implies that the spanning set of Br(Bn) is abasis and that Br(Bn) is free of rank f (n). In particular, φ : Br(Bn) → SBr(Dn+1) is anisomorphism and Theorem 3.1.2 is proved.

3.5 CELLULARITY 73

3.5 Cellularity

Recall the definition of cellular algebra in Section 1.4.

Theorem 3.5.1. There is a cellular datum for Br(Bn)⊗Z[δ±1] R if R is an integral domainin which 2 and δ are invertible elements.

Proof. Let R be as indicated and write A = Br(Bn)⊗

Z[δ±1] R. We introduce a quadruple(Λ, T, C ,∗) and prove that it is a cell datum for A. The map ∗ on A will be the natural anti-involution ·op on A over R. By Proposition 1.6.3, this map is an R-linear anti-involution ofA.

By Theorems 3.4.8 and 3.1.2, the Brauer algebra A over R has a basis consisting ofthe elements of Theorem 3.4.8.

For t ∈ {0, . . . , bn/2c}, let C∧t =ψφ(C(1)t ) and Ct =

D

ψ(R2),ψφ(C(2)t )E

∼=W (An−2t)⊂W (Dn+1), and put Y = C∧t or Ct . As Y is a Weyl group with irreducible factors of typeB or A and the coefficient ring R satisfies the conditions of [24, Theorem 1.1], we con-clude from [24, Corollary 3.2] that the group ring R[Y ] is a cellular subalgebra of A. Let(ΛY , TY , CY ,∗Y ) be the corresponding cell datum for R[Y ]. By [24, Section 3], ∗Y is themap ·op on R[Y ].

The set Λ is defined as the union of Λ1 and Λ2, where Λ1 = {t}[ n

2]

t=0 and Λ2 =

{(t,θ)}[ n+1

2]

t=1 . Here θ is used to guarantee that the two sets are disjoint. A set of t un-ordered pairs in {1, . . . , n+1} is called an admissible t-set in {1, . . . , n+1} if no two pairshave a common number. We denote the set of all admissible t-sets of {1, . . . , n+ 1} byUn+1

t . A decorated pair in {1, . . . , n+1} is a triple {i, j,+} or {i, j,−} with 1≤ i, j ≤ n+1with ± for decorations. A decorated admissible t-set in {1, . . . , n+1} is an admissible t-setin {1, . . . , n+ 1} with each pair being decorated. We denote all decorated admissible t-sets in {1, . . . , n+1} by U∧n+1

t . The set of all decorated admissible t-sets in {1, . . . , n+1}without 1 appearing in any pair is denoted by U |∧n+1

t . We view Un+1t as the subset of

U∧n+1t of all admissible t-sets all of whose pairs are decorated by −. For each t ∈ Λ1, we

define the associated finite set to be

T (t) = {(u, v) | u ∈ U |∧n+1t , v ∈ TC∧t

}.

For each (t,θ), we define the associated finite set to be

T ((t,θ)) = {(u, v) | u ∈ Un+1t , v ∈ TCt

}.

By Theorem 3.2.8 and (C1) for R(Y ), there exists a map

D :∐

λ∈Λ

T (λ)× T (λ)→ BrMD(Bn),

where the diagram of D((u1, v1), (u2, v2)) is given as follows: the top horizontal strandsare strands between pairs in u1 and decorated for +; the bottom horizontal strands aresimilarly given by u2; the free (decorated) vertical strands (and multiplied by ξδ or not)are given by CY (v1, v2), multiplied by θδ−1 if λ ∈ Λ2. We see that D is an injective mapand that its image is a basis of BrMD(Bn). Therefore we define C = φ−1ψ−1D. Thepartial order on Λ is given by

74 TYPE Bn

• λ1 ≺ λ2 if λ1, λ2 ∈ Λ1 and, as integers, λ1 > λ2,

• λ1 ≺ λ2 if λ1 = (t1,θ), λ2 = (t2,θ) ∈ Λ2 and t1 > t2,

• λ1 ≺ λ2 if λ1 = t1 ∈ Λ1 and λ2 = (t2,θ) ∈ Λ2 and t1 ≥ t2.

The partial order is illustrated in the following Hasse diagram, in which a � b is equiv-alent to the existence of a directed path from a to b. In the diagram, the top row andthe bottom row end at n

2and ( n

2,θ), respectively, when n is even; the top row and the

bottom row end at n−12

and ( n+12

,θ), respectively, when n is odd.

0 // 1 //

��

2

��

// 3

��

// · · ·

��(1,θ) // (2,θ) // (3,θ) // · · ·

In other words, we inherit the cellular structure of Br(Dn+1) in [9, section 6]. By Theo-rem 3.1.2 and Theorem 3.2.8, the quadruple (Λ, T, C ,∗) satisfies (C1). From the diagramrepresentation of BrMD(Bn) described in Theorem 3.2.8 and (C2) for R[Y ], the quadruple(Λ, T, C ,∗) satisfies (C2) with ∗ = ·op. It remains to check condition (C3) for (Λ, T, C ,∗).To this end we just need to consider riC((u1, v1), (u2, v2)) and eiC((u1, v1), (u2, v2)). Thecorrectness of the required expression can be proved by a case-by-case check using (C3)of R[Y ] and either the lemmas of Section 3.4 (in which case the check involves the obser-vation that φ−1ψ−1(θ(T 0

n+1 ∩ T=n+1)) is a monoid ideal inside M , and that multiplication

by a generator always brings a member of W f (i)t W to a member W f ( j)t ′ W for some j witht ′ ≥ t), or an argument using the diagram representation of BrMD(Bn).

We conclude that (Λ, T, C ,∗) is a cell datum for Br(Bn)⊗Z[δ±1] R.

Example 3.5.2. When n= 2, and 3, the Hasse diagrams in the above proof are

0 // 1

��(1,θ)

and 0 // 1

��(1,θ) // (2,θ)

,

respectively. The cardinalities of the parts (spanning Aλ mod A≺λ) of the basis associatedto each node λ in these two diagrams are

0 1 (1,θ)8 9 9

0 1 (1,θ) (2,θ)48 144 72 9

respectively. For n = 3, the correspondence with the elements of F in (3.1.1) (and theconnected decomposition of Br(B3) given there) is as follows: 0 corresponds to 1 (its48 elements are those of W (B3)), 1 corresponds to the element e2, the element (1,θ)corresponds to all other elements of F but the final one, and (2,θ) corresponds to e0e2.

The sum⊕

1≤t≤(n+1)/2 A(t,θ) is an ideal of A = Br(Bn) isomorphic to the ideal ofBr(An)⊗ R spanned by all diagrams with a horizontal strand.

3.5 CELLULARITY 75

Remark 3.5.3. In [29], König and Xi proved that Brauer algebras of type A are inflationcellular algebras, and also in [2], Bowman proved that the Brauer algebras found in [14]are cellularly stratified algebras (a stronger version of inflation cellular algebras). Bothkinds of algebra have totally ordered sets Λ associated to the cellular structures. Fromthe Hasse diagrams, it can be read off that, in our case, Λ is totally ordered when n = 2,3 and not totally ordered when n> 3. These observations explain why we have not beenable to use [29] for a cellularity proof. A proof similar to the one in [2] gives that thealgebras Br(B2)⊗Z[δ±1] R and Br(B3)⊗Z[δ±1] R are cellularly stratified algebras if R is anintegral domain in which 2 and δ are invertible elements.

76 TYPE Bn

4TYPE F4

4.1 Introduction

In Chapter 1, we gave the definition of Brauer algebras of simply laced types, includingtype E6. Tits ([39]) described how to obtain the Coxeter group of type F4 as the fixedsubgroup of the Coxeter group of type E6 under a diagram automorphism. Here we willprove a similar relation between Br(F4) and Br(E6). This study continues the last twochapters. It turns out that the presentation by generators and relations obtainable fromthe Dynkin diagram of type F4 in the same way as was done before for type B and Cleads to an algebra isomorphism to a subalgebra of Br(E6). First we recall the definitionof Br(F4) in Definition 1.6.1. The defining relations (1.6.10)–(1.6.17) relevant for gen-erators r2, r3, e2, e3, can be found in Br(C2) in Chapter 2 for generators r1, r0, e1, e0,respectively, and Br(B2) in Chapter 3 for generators r0, r1, e0, e1, respectively. Note thatthese relations are not symmetric in 2 and 3. Their relations are fully determined by theDynkin diagram in the sense that all relations depend only on the vertices and bonds ofthe Dynkin diagram and the lengths of their roots.It is well known that the Coxeter group W (F4) of type F4, can be obtained as the subgroupfrom the Coxeter group W (E6) of type E6, of elements invariant under the automorphismof W (E6) determined by the diagram automorphism σ = (1,6)(3, 5) given as a permuta-tion of the generators of W (E6) which are labeled and indicated in Figure 4.1.

The action σ can be extended to an automorphism of the Brauer algebra Br(E6) oftype E6 by acting on the Temperley-Lieb generators Ei ([38]) by the same permutationof indices as on the Weyl group generators. We denote SBrM(E6) the submonoid of σ-invariant monomials in BrM(E6) and SBr(E6) the linear span of SBrM(E6) over Z[δ±1].The main theorem of this paper is the following. In order to avoid confusion with theabove generators, the generators of Br(E6) have been capitalized.

Theorem 4.1.1. There is an algebra isomorphism

φ : Br(F4)−→ SBr(E6)

determined by φ(r1) = R1R6, φ(r2) = R3R5, φ(r3) = R4, φ(r4) = R2, and φ(e1) = E1E6,φ(e2) = E3E5, φ(e3) = E4, φ(e4) = E2. Furthermore, the algebra Br(F4) is free over Z[δ±1]of rank 14985.

78 TYPE F4

1

2

3 4 5 6

<

1 2 3 4

E6

F4

Figure 4.1: Dynkin diagrams of E6 and F4

The proof of the theorem is given at the end of Section 4.5. Furthermore, in theSection 4.6 we will prove that the algebra Br(F4)⊗R is cellularly stratified for the field Rwith 2 and 3 being invertible in R with δ specified to a nonzero value.

4.2 Basic properties of Br(F4)

By the properties of Br(B3) in Chapter 3 and of Br(C3) in Chapter 2, we have morerelations between {r2, r3, e2, e3} than given in Definition 1.6.1, such as Lemma 1.6.2.

Proposition 4.2.1. The map φ determined on generators as in Theorem 4.1.1 induces anZ[δ±1]-algebra homomorphism from Br(F4) to Br(E6).

Proof. The relations in Definition 1.6.1 still hold when the generators are replaced bytheir images under φ. The difficult ones are (1.6.10)–(1.6.17), which have been verifiedfor the homomorphism from Br(C2) to Br(A3) in Lemma 2.2.4.

To distinguish them from the generators of Br(F4), we denote the generators ofBr(C3) (◦

2◦1

< ◦0) as in Chapter 2 by {r ′i , e′i}

2i=0 and the generators of Br(B3)

(◦2

◦1

> ◦0) in Chapter 3 by {r ′′i , e′′i }

2i=0. By checking their defining relations,

we have the following proposition.

Proposition 4.2.2. There are injective algebra homomorphisms

φ1 : Br(C3)→ Br(F4)

andφ2 : Br(B3)→ Br(F4),

defined on generators as follows.

φ1(r′i ) = r3−i , φ1(e

′i) = e3−i , for 0≤ i ≤ 2,

φ2(r′′i ) = r2+i , φ2(e

′′i ) = e2+i , for 0≤ i ≤ 2.

4.3 THE ROOT SYSTEM OF TYPE F4 79

Proof. Just checking the defining relations of Br(B3), Br(C3), we find that φ1 and φ2are algebra homomorphisms. We see that φφ1(Br(C3)) is contained in the subalgebraof Br(E6) generated by {R1, E1} ∪ {Ri , Ei}6i=3, which is isomorphic to Br(A5) accordingto [17]. By Theorem 2.1.1 in Chapter 2 and the behavior of φφ1 on generators, thehomomorphism φφ1 coincides with the embedding of Br(C3) into Br(A5). Similarly, dueto [17] and Theorem 3.1.2 in Chapter 3, the homomorphism φφ2 coincides with theembedding of Br(B3) into Br(D4); by application of these two theorems, it follows thatφ1 and φ2 are injective.

4.3 The root system of type F4

In this section, we give the definition and description of admissible sets of type F4 andstudy some of their basic properties.Let {βi}4i=1 be simple roots of W (F4). They can be realized in R4 as

β1 =ε1 − ε2 − ε3 − ε4

2, β2 = ε2,

β3 = ε3 − ε2, β4 = ε4 − ε3,

with {εi}4i=1 being the standard orthonormal basis of R4. The set

Ψ+ =�

ε1 ± ε2 ± ε3 ± ε4

2

�

∪ {εi}4i=1 ∪ {ε j ± εi}1≤i< j≤4

of cardinality 24 is the set of positive roots of root system Ψ of W (F4) having β1, . . ., β4as simple roots. We call a vector β ∈ Ψ+ a short root if its Euclidean length is 1, a longroot if its Euclidean length is

p2.

Let {αi}6i=1 be simple roots of W (E6). The {αi}6i=1 span a linear space over R of dimension6. As in Section 1.6, the p is given as follows:p(α1) = β1, p(α6) = β1, p(α3) = β2, p(α5) = β2, p(α4) = β3, p(α2) = β4.Now p(R6) is the σ-invariant space of R6, where σ is the linear transformation of R6

determined by:σ(α1) = α6, σ(α6) = α1, σ(α3) = α5, σ(α5) = α3, σ(α4) = α4, σ(α2) = α2.Let Φ ⊂ R6 be the root system of type E6 with simple roots {αi}6i=1, and Φ+ the positiveroots of Φ. Recall the Definition 1.6.4. By computation, we get the following proposition.

Proposition 4.3.1. There is a one-to-one correspondence between σ-invariant admissibleroot sets of type E6 and the admissible root sets of type F4. The collections of all admissiblesets can be partitioned into six W (F4)-orbits given by the following representatives.

(I) ;,

(II) {ε4 − ε3},

(III) { ε1−ε2−ε3−ε4

2},

(IV) {ε3 − ε2,ε3 + ε2},

(V) { ε1−ε2−ε3−ε4

2,ε3 − ε2,ε4 + ε1},

80 TYPE F4

(VI) {ε3 − ε2,ε3 + ε2,ε4 + ε1,ε4 − ε1}.

Furthermore, their cardinalities are 1, 12, 12, 18, 36, 3, respectively.

Proof. In the admissible root sets of type E6, there are four W (E6)-orbits, which are theorbits of ;, {α2}, {α3, α5}, {α2, α3, α5, 2α4 + α2 + α3 + α5} of respective sizes. TheW (E6)-orbits are easily seen to possess 1 ((I)), 12 ((II)), 30 ((III), (IV)), 39 ((V), (VI))σ-invariant admissible root sets which can be decomposed into W (F4)-orbits with repre-sentatives (I), (II), (III), (IV),(V), (VI) as listed in the proposition.

Recall (1.6.30). Analogous to the argument in Lemma 3.3.5, the following lemmacan be obtained by checking case by case listed in Proposition 4.3.1.

Lemma 4.3.2. Let γ1, γ2 ∈ Ψ+ and γ1 orthogonal to γ2 such that {γ1, γ2} is a subset ofsome admissible root set. Then

eγ1eγ2= eγ2

eγ1.

If X ⊂Ψ+ is a subset of some admissible root set, then by the lemma we can define

eX = Πβ∈X eβ . (4.3.1)

Thanks to an argument similar to Lemma 3.3.6, the following lemma holds.

Lemma 4.3.3. Let X ⊂Ψ+ be a mutually orthogonal root set. If X cl exists, then

eX cl = δ#(X cl\X )eX .

4.4 An upper bound for the rank

We introduce notation for the following admissible sets of type F4 corresponding to thoselisted in Proposition 4.3.1.

X0 = ;X1 = {β4}X2 = {β1}X3 = {β3,β3 + 2β2}X4 = {β1,β3}cl

X5 = {β3,β3 + 2β2,β3 + 2β2 + 2β1}cl

We define the subgroups Ni , Ci , Ai for i = 0, . . . , 5 and Ni is the subgroup of W (F4)generated by Ai and Ci as follows.

(I) N0 =W (F4), C0 = N0, A0 = {1},

(II) N1 =¬

r1, r2, rε4+ε3, r4

¶

, C1 =¬

r1, r2, rε4+ε3

¶

, A1 =

r4�

,

(III) N2 =¬

r1, r3, r4, r(ε1+ε2+ε3−ε4)/2

¶

, C2 =

r3, r4�

,

A2 =¬

r1, r(ε1+ε2+ε3−ε4)/2, r(ε1+ε2−ε3+ε4)/2, r(ε1−ε2+ε3+ε4)/2

¶

,

(IV) N3 =¬

r2, r3, rε4, rε4−ε1

¶

, C3 =¬

rε4−ε1

¶

, A3 =¬

r2, r3, rε4, rε1

¶

,

4.4 AN UPPER BOUND FOR THE RANK 81

(V) N4 =¬

r3, r1, r(ε1+ε2+ε3−ε4)/2, r(ε1−ε2+ε3+ε4)/2

¶

, C4 = {1}, A4 = N4,

(VI) N5 =¬

r1, r2, r3, rε4, rε1

¶

, C5 = {1}, A5 = N5.

The structures of these subgroups are determined below.

Lemma 4.4.1. For i = 0, . . . , 5, the subgroups Ni , Ai , Ci satisfies the following isomorphism.

(I) N0∼=W (F4), A0

∼= {1}, C0∼= N0,

(II) N1∼=W (B3)×W (A1), C1

∼=W (B3), A2∼=W (A1),

(III) N2∼=W (B3)×W (A1), C2

∼=W (A2), A2∼=W (A1)4,

(IV) N3∼=W (B2)2, C3

∼=W (A1), A3∼=W (B2)×W (A1)2,

(V) A4 = N4∼=W (B2)×W (A1)2, C4 = {1},

(VI) A5 = N5∼=W (B3)×W (B2), C5 = {1}.

Proof. We do not give the full proof but restrict to the case i = 1. It can be checked that

ε3 + ε4,β1�

=−1 and

ε3 + ε4,β2�

= 0, and {ε3+ ε4,β1,β2} are linearly independent,hence C1

∼=W (B3). Since each element of {ε3+ ε4,β1,β2} is orthogonal to β4, so we getthat N1

∼=W (B3)×W (A1).

When we consider these groups in BrM(F4), the following lemma can be obtained.

Lemma 4.4.2. For i = 0, . . ., 5, the following holds.

(I) The group Ni is the normalizer of X i in W (F4).

(II) The group Ni is the semidirect product of Ai and Ci , with Ai normalized by Ci .

(III) For x ∈ Ai , xeX i= eX i

.

(IV) For x ∈ Ci , xeX i= eX i

x.

Proof. Clearly, Ni normalizes X i , so Ni ≤ N(X i) (the normalizer of X i in W (F4)), and theequality follows from Lagrange’s Theorem by verification in the table below. Here #Niis known from Lemma 4.4.1, and the lengths of W (F4)-orbits are given in Proposition4.3.1. Therefore the first claim hold. The proof of the remaining conclusions is in thearguments in proofs of Lemma 2.5.4 and Lemma 3.4.4.

For i ≤ 5, suppose Di is a set of left coset representatives of Ni in W (F4). In the tablebelow the product of the three entries in each row is equal to 1152.

i #Di #Ci #Ai

0 1 1152 11 12 48 22 12 6 163 18 2 324 36 1 325 3 1 384

82 TYPE F4

We find that for the Brauer monoid action of some monomial φ(a) for a ∈ BrM(F4),the admissible root sets p(φ(a);) and p(φ(a)op;) belong to different W (F4)-orbits; forexample p(φ(e2e3);) = {β2}, and p(φ(e2e3)op;) = {β3,β2 + β3}. Some more groups asfollows are needed. Let

N L6 =

¬

r2, r3, rε4, rε4−ε1

¶

, (4.4.1)

NR6 =

¬

r2, rε3, rε4

, rε4−ε1

¶

, (4.4.2)

C6 =¬

rε4−ε1

¶

, (4.4.3)

N8 =¬

r2, r4, rε1, rε3

¶

. (4.4.4)

Additionally, we choose DL6 , DR

6 , and D8 to be sets of left coset representatives of N L6 , NR

6 ,and N8 in W (F4), respectively.LetN L

7 = NR6 , NR

7 = N L6 , DL

7 = DR6 ,

DR7 = DL

6 , C7 = C6,N L

9 = N5, NR9 = N4, DL

9 = D5, DR9 = D4,

N L10 = N4, NR

10 = N5, DL10 = D4, DR

10 = D5.In view of Lemma 3.4.7, the following lemma holds.

Lemma 4.4.3. The above groups satisfy the following properties.

(I) N L6∼=W (B2)×W (B2), NR

6∼=W (B2)×W (A1)2, C6

∼=W (A1), N8∼=W (B2)×W (A1)2.

(II) For each a ∈ N L6 , b ∈ NR

6 , there exists some c, d ∈ C6 such that ae3e2 = e3e2c ande3e2 b = de3e2, respectively.

(III) For each a ∈ N8 we have ae4r3e2e3e4 = e4r3e2e3e4.

(IV) For each a ∈ N L9 and b ∈ NR

9 , we have ae3e2e1e3 = e3e2e1e3 and e3e2e1e3 b =e3e2e1e3.

Theorem 4.4.4. Up to some power of δ, each monomial in BrM(F4) can be written in oneof the following forms.

(I) ueX ivw, u ∈ Di , w ∈ D−1

i , v ∈ Ci , 0≤ i ≤ 5.

(II) ue3e2vw, u ∈ DL6 , w ∈ (DR

6)−1, v ∈ C6.

(III) ue2e3vw, u ∈ DL7 , w ∈ (DR

7)−1, v ∈ C7.

(IV) ue4r3e2e3e4w, u ∈ D8, w ∈ D−18 .

(V) ue3e2e1e3w, u ∈ DL9 , w ∈ (DR

9)−1.

(VI) ue3e1e2e3w, u ∈ DL10, w ∈ (DR

10)−1.

Proof. From Lemma 3.2.5, (e4r3e2e3e4)op = e4r3e2e3e4. In view Proposition 1.6.3, toprove this theorem it suffices to prove the claim that the result of a left multiplication byeach ri and eβ for β ∈ Ψ+ at the left of each element of S can be written as in (I)–(VI),where

S = {eX i}5i=0 ∪ {e3e2, e2e3, e4r3e2e3e4, e3e2e1e3, e3e1e2e3}.

4.5 φ(Br(F4)) IN Br(E6) 83

According to Lemma 4.4.2 and Lemma 4.4.3, the above holds for {ri}4i=1.By Proposition 4.2.2, and special cases Br(C3) in Chapter 2 and Br(B3) in Chapter 3, wehave that

φ1(Br(C3)) =⊕

s∈S1

Z[δ±1]W (C3)sW (C3),

φ2(Br(B3)) =⊕

s∈S2

Z[δ±1]W (B3)sW (B3),

WhereS1 = {1, e3, eX2

, e3e2, e2e3, eX3, eX4

, e3e2e1e3, e1e3e2e3, eX5}

S2 = {1, eX1, e2, e3e2, e2e3, eX3

, e4e∗4, e4e2, e4r3e2e3e4}

with e∗4 = r3r2r3e4r3r2r3. It can be seen that each element of S1 and S2 is in S or conjugateto some element of S under W (F4). Therefore the proof can be reduced to cases of Br(B3)and Br(C3), which can be found in Sections 2.5 and 3.4.

As a consequence, we obtain some information about the rank of Br(F4) over Z[δ±1].

Corollary 4.4.5. As a Z[δ±1]-algebra, the algebra Br(F4) is spanned by 14985 elements.

Proof. For i = 6, . . ., 10, the following holds.

set cardinality set cardinalityDL

6 , DR7 18 DR

6 , DL7 36

C6, C7 2 D8 36DL

9 , DR10 3 DR

9 , DL10 36

By Theorem 4.4.4 and numerical information from the above two tables, the algebraBr(F4) over Z[δ±1] has rank at most

5∑

i=0

(#Di)2#Ci + 2#DL

6 #DR6#C6 +#D2

8 + 2#DL9 #DR

9 = 14985.

4.5 φ(Br(F4)) in Br(E6)

Recall Theorem 1.5.11. We give the normal forms of Br(E6) in view of σ.Consider our σ and [17, Table 3] or Table 1.2 and let

Y ∈ Y = {;, {2}, {1, 6}, {2, 3,5}}.

Obviously, each element of Y is σ-invariant. Each monomial a in BrM(E6) can beuniquely written as δiaB EY haop

B′ for some i ∈ Z and h ∈ W (MY ), where W (MY ) isthe group of invertible elements in EY W (M)EY , and B = a;, B

′= ;a, aB ∈ BrM(E6),

aopB′ ∈ BrM(E6) and

(i) a;= aB;= aBBY , ;a = ;aopB′ = BY aop

B′ ,

84 TYPE F4


If a = δiaB EY haopB′ is σ-invariant, since σ(Y ) = Y , it is equivalent to say that aB aop

B′ areσ-invariant. Now we can obtain the following corollary immediately.

Corollary 4.5.1. Let B, Y , B′ be as above. Then a = aB EY haopB′ is σ-invariant monomial in

SBrM(E6) if and only if

(i) the sets B, B′are σ-invariant,

(ii) the element h ∈W (MY ) is σ-invariant.

In type E6, we see that E2E4E5{α6}= {α2}, E1E3{α4,α6}= {α1,α6}. Then

W (M{2}) = E2E4E5W (M{6})E5E4E2,

W (M{1,6}) = E1E3W (M{4,6})E3E1.

Let Ei = δ−1Ei . By the 6th column listed in [17, Table 3], the group W (M{2}) ∼=W (A5) has generators R1 E2, R3 E2, R5 E2, R6 E2, E2E4R3E5E4 E2 and the Dynkin diagram ofW (M{2}) is corresponding to the subdiagram of E6 by deleting the node 2 and the edgebetween 2 and 4 with E2E4R3E5E4 E2 corresponding to node 4, Ri E2 corresponding tonode i for i = 1, 3, 5, 6; the group W (M{1,6}) ∼= W (A2) has generators R2 E1 E6, R4 E1 E6and the Dynkin diagram of W (M{1,6}) is the subdiagram of E6 of nodes 2, 4 and the edgebetween them. When σ acts on the generators of W (M{2}) and W (M{1,6}), we have that

σ(E2E4R3E5E4 E2) = E2E4R3E5E4 E2,

σ(Ri E2) = Rσ(i) E2, (i = 1,3, 5,6),σ(R j E1 E6) = R j E1 E6, ( j = 2, 4),

which implies that the σ-action on those two groups is determined by the σ-action onthe subdigrams of E6 described in the above. Let W (M{2})σ and W (M{1,6})σ be the σ-invariant subgroups of W (M{2}) and W (M{1,6}), respectively. Hence W (M{2})σ ∼=W (B3),and W (M{1,6})σ = W (M{1,6}) ∼= W (A2). This conclusion can be summarized in the tablebelow with the second column from GAP [23] code.

Y #((W (E6)BY )σ) MY MσY

; 1 E6 F42 12 A5 B3

1, 6 30 A2 A22, 3, 5 39 ; ;

Corollary 4.5.2. The algebra SBr(E6) is free over Z[δ±1] with rank

1152+ 122 × 48+ 302 × 6+ 392 = 14985.

Proof. By Corollary 4.5.1, we have

rk(SBr(E6)) = ΣY∈Y (#((W(E6)BY)σ))2#W(MY)

σ.

The freeness holds because of the freeness of Br(E6). Hence the corollary holds.

4.6 CELLULARITY 85

The following can be checked by computation by using Theorem 4.4.4.

Lemma 4.5.3. Let KY = {a ∈ BrM(E6) | σ(a) = a, a; ∈ W (E6)BY } for Y ∈ Y . ThenKY = {φ(b) | b ∈ BrM(F4), φ(b); ∈W (E6)BY }.

Proof. If Y = {2}, then {φ(b) | b ∈ BrM(F4), φ(b); ∈ W (E6)BY } are the elements in (I)for i = 1 in Theorem 4.4.4, which has cardinality 122 × 48, and E2φ(C1)E2 = W (MY )σ

(in the proof of Corollary 4.5.1), and the φ(W (F4))-orbit of {α2} is σ-invariant elementsin W (E6)BY . Hence the lemma holds for Y = {2}.If Y = {1,6}, then {φ(b) | b ∈ BrM(F4), φ(b); ∈ W (E6)BY } are those monomials in(I) for i = 2 and i = 3, (II), (III) and (IV) in Theorem 4.4.4. Analogous to the aboveargument for Y = {2}, we see that the image of those monomials under φ correspondto different normal forms in Theorem 1.5.11 for BrM(E6) up to some powers of δ, andthose monomials of in (I) for i = 2 and i = 3, (II), (III) and (IV) in Theorem 4.4.4 havecardinality 122 × 6+ 182 × 2+ 18× 36× 2+ 18× 36× 2+ 362 = 302 × 6. Hence thelemma holds for Y = {1,6}.If Y = {2, 3,5}, then {φ(b) | b ∈ BrM(F4), φ(b); ∈W (E6)BY } are those monomials in (I)for i = 4 and i = 5, (V), and (VI) in Theorem 4.4.4. Similarly, we see those monomialsare corresponding to different normal forms in Theorem 1.5.11 for BrM(E6) up to somepowers of δ, and they have cardinality 362+32+36×3+36×3= 392. Hence the lemmaholds for Y = {2, 3,5}.

Now, we can give the proof of our main Theorem 4.1.1.

Proof. Proposition 4.2.1 implies that φ is a homomorphism. Corollary 4.4.5 indicatesthat φ(Br(F4)) has rank at most 14985. Corollary 4.5.2 implies that SBr(E6) has rank14985. Lemma 4.5.3 indicates that φ has image SBr(E6); therefore φ is surjective. Thehomomorphism φ is an isomorphism and Br(F4) is free over Z[δ±1] of rank 14985 sincethe freeness of Br(E6).

4.6 Cellularity

Recall the definition of cellular algebra and cellularly stratified algebra from Section 1.4in Chapter 1. As in [2], the following theorem can be obtained.

Theorem 4.6.1. Let R be a field with 2 and 3 being invertible in R with δ given a nonzerovalue. Then the algebra Br(R, F4) = Br(F4)⊗Z[δ±1] R is a cellularly stratified algebra over R.

Proof. To prove this, it suffices to prove it for SBr(R, E6) = SBr(E6)⊗Z[δ±1]R because of theisomorphism in Theorem 4.1.1. Let Z0 ⊂ Z1 ⊂ Z2 ⊂ Z3 be a σ-invariant and admissibleroot set sequence of type E6, where Z0 = ;, Z1 = {α2}, Z2 = {α2,α2 + α3 + α5 + 2α4},Z3 = {α2,α3,α5,α2 +α3 +α5 + 2α4}. As E2E4E5E3{α1,α6}= Z2, we have ht(Z2) = 0.For 0 ≤ i ≤ 3, let BZi

be the group algebras of WZ0= W (Mσ

; ), WZ1= W (Mσ

{2}), WZ2=

E2E4E5E3W (Mσ{1,6})E3E5E4E2, WZ3

=W (Mσ{2,3,5}) over R, respectively, whose group rings

over R are cellular algebras due to [24, Theorem 1.1].Then each monomial a in SBrM(E6) can be uniquely written as δiaZi ,B EY haop

Zi ,B′for some

i ∈ {0,1, 2,3} and h ∈ WZi, where B = a;, B

′= ;a are σ-invariant, aZi ,B ∈ SBrM(E6),

aopZi ,B′∈ SBrM(E6) and

86 TYPE F4

(i) a;= aZi ,B;= aB Zi , ;a = ;aopZi ,B′= Zia

opB′ ,

(ii) ht(B) =ht(aZi ,B), ht(B′) =ht(aopZi ,B′).

For each Zi , let VZibe a linear space over R with basis uZi ,B where B ∈ W (E6)Zi and

σ(B) = B. and let ϕZibe a bilinear map defined as

VZi⊗R VZi

−→ BZi

ϕZi(uZi ,B, uZi ,B′) = aop

Zi ,BaZi ,B′ , if Zi = aop

Zi ,BaZi ,B′;,

ϕZi(uZi ,B, uZi ,B′) = 0, if Zi $ aop

Zi ,BaZi ,B′;.

We first prove that ϕZiis well defined. As aop

Zi ,B= Zi ,we find Zi ⊂ aop

Zi ,BaZi ,B′;; similarly

Zi ⊂ ;aopZi ,B

aZi ,B′ . If Zi = aopZi ,B

aZi ,B′;, this indicates that Zi = ;aopZi ,B

aZi ,B′ and that aopZi ,B

aZi ,B′

will be in WZiup to some power of δ. Therefore our ϕZi

is well defined. Observe that

(aopZi ,B

aZi ,B′)op = aop

Zi ,B′(aop

Zi ,B)op = aop

Zi ,B′aZi ,B,

so (ϕZi(uZi ,B, uZi ,B′))

op = ϕZi(uZi ,B′ , uZi ,B). By linear extension, we find (ϕZi

(u, v))op =ϕZi(v, u), for u,v ∈ VZi

. By the proof of Lemma 4.5.3, the algebra SBr(R, E6) is an it-erated inflation of the cellular algebra BZi

along vector space VZifor 0 ≤ i ≤ 3, namely

SBr(R, E6) satisfies (1) of a cellularly stratified algebra in Section 1.4. We take eZi= EZi

=uZi ,Zi

⊗ uZi ,Zi⊗ 1BZi

, where 1BZi= δ−#Zi EZi

. Because EZiEZ j= δ#Zi EZ j

for Zi ⊂ Z j , henceconditions (2) and (3) follows since that Zi > Z j means Zi $ Z j . Finally, SBr(R, E6) is acellularly stratified algebra.

5TYPE In

2

5.1 Introduction

In [33], Mühlherr described how to get the Coxeter group of type In2 from a diagram

operation as a subgroup of the Coxeter group of type An−1. Here we will apply a similarapproach to Mühlherr’s on Br(An−1), to get Br(In

2). We first give the definition of Br(In2)

for n ≥ 5 by generators and relations. The definitions depend on the parity of n and soare divided between n= 2m and n= 2m− 1.

Definition 5.1.1. The Brauer algebra of type I2m2 for m ∈ N>2, denoted by Br(I2m

2 ), is aunital associative Z[δ±1]-algebra generated by r0, r1, e0 and e1 subject to the followingrelations and a set Θ ⊂ N consisting of κi , η j , ξ j θ j , where i = 0, 1, j = 1, . . ., [m/2].Symbols [r0r1 · · · ]t and [r1r0 · · · ]t stand for words of length t with r0 and r1 iterated.

r2i = 1 (5.1.1)

riei = ei ri = ei (5.1.2)

e2i = δκi ei (5.1.3)

e1e0e1 = δe1 (5.1.4)

e0[r1r0 · · · ]2m−1 = [r1r0 · · · ]2m−1e0, (5.1.5)

e1[r0r1 · · · ]2m−1 = e1, (5.1.6)�

r0r1 · · ·�

2m−1 e1 = e1, (5.1.7)

e0[r1r0 · · · ]2ke1 = δθk e0e1, 0≤ k ≤ [m/2] (5.1.8)

e1[r0r1 · · · ]2ke0 = δθk e1e0, 0≤ k ≤ [m/2] (5.1.9)

e1[r0r1 · · · ]2k−1e1 = δηk e1, 0≤ k ≤ [m/2] (5.1.10)�

r1r0 · · ·�

2m = [r0r1 · · · ]2m (5.1.11)

88 TYPE In2

and when 2k ≤ m, let l = lcm(k, m),

e0[r1r0 · · · ]2k−1e0 = δξk[r1r0 · · · ]2m−1e0,

l

m,

l

kodd (5.1.12)

e0[r1r0 · · · ]2k−1e0 = δξk e0,

l

meven (5.1.13)

e0[r1r0 · · · ]2k−1e0 = δξk e0e1e0,

l

modd,

l

keven (5.1.14)

The submonoid of the multiplicative monoid of Br(I2m2 ) generated by δ, {ri}1i=0 and

{ei}1i=0 is denoted by BrM(I2m2 ). This is the monoid of monomials in Br(I2m

2 ).

Definition 5.1.2. The Brauer algebra of type I2m−12 for m ∈ N>2, denoted by Br(I2m−1

2 ), isa unital associative Z[δ±1]-algebra generated by r0, r1, e0 and e1 subject to the followingrelations and a set Θ ∈ N consisting of κi , ξ j with j = 1, . . ., m, i = 0, 1, κ0 = κ1.

r2i = 1 (5.1.15)

riei = ei ri = ei (5.1.16)

e2i = δκi ei (5.1.17)

�

r0r1 · · ·�

2m−2 e0 = e1[r0r1 · · · ]2m−2, (5.1.18)

e0[r1r0 · · · ]2k−1e0 = δξk e0, 0< k < m (5.1.19)�

r1r0 · · ·�

2m−1 = [r0r1 · · · ]2m−1 (5.1.20)

The submonoid of the multiplicative monoid of Br(I2m−12 ) generated by δ, r0, r1, e0,

e1 is denoted by BrM(I2m−12 ). This is the monoid of monomials in Br(I2m−1

2 ).

It is well known from [33] that the Coxeter group of type In2, denoted by W (In

2) canbe realized as a subgroup from the Coxeter group of type An−1 denoted by W (An−1) withrespect to an admissible partition (Definition 1.6.8) of the nodes of the Coxeter diagramof type An−1. The partition is the division of the nodes of type An−1 into two subsets A, Bwhich are even nodes, and odd nodes, respectively. Figure 5.1 denotes the diagram fortype A5 and I6

2 in such a way that the nodes from the same parts of A5 of the admissiblepartition for I6

2 occur at the same level. The main theorem of this chapter is the following.

1

2

3

4

5

1,3,5 2,4

Figure 5.1: Partition for I62 in A5

5.2 AN INTERESTING ELEMENTARY PROBLEM 89

Theorem 5.1.3. For n> 4, there is an algebra isomorphism

φ : Br(In2)−→ Br(An−1)

determined by φ(r0) =∏0<i<n

i even Ri , φ(r1) =∏0<i<n

i odd Ri , φ(e0) =∏0<i<n

i even Ei and φ(e1) =∏0<i<n

i odd Ri when each parameter in Θ takes special value in N to make φ an algebra homo-morphism. Furthermore,

rankZ[δ±1]Br(In2) =

¨

2n+ n2, if n is odd,2n+ 3

2n2, if n is even.

We will compare Br(I62) and Br(G2) in Remark 7.5.2.

5.2 An interesting elementary problem

Suppose that k, m ∈ N are such that 1 < 2k ≤ m. There is a box in the x , y plane R2

fixed by four lines x = 1, x = 2m, y = 2k− 12, and y =− 1

2. Imagine you have a particle,

which starts to move from (1, 2k − 1) with slope −1; when it touches the bottom (thetop), it will be reflected with the bottom (the top) as a mirror; but when it touches theright (left) wall, it first goes down (up) 1 unit vertically, if it comes at the wall from thetop (bottom), and continues its path with the wall as the mirror; it stops if it reachesthe points (1, 0), (2m, 0), (1,2k − 1), or (2m, 2k − 1). For different values of k, m, theproblem is to decide at which point the particle stops. One example is Figure 5.2, whenm= 5, k = 2.

1 2 3 4 5 6 7 8 9 10

1

2

3

72

−12

Figure 5.2: case for m= 5, k = 2

To solve the problem, we unfold its path by "penetrating" the walls, which means thatwhen the particle touches the right wall for the first time, we change the vertical step intoone move of slope−1 when coming from the top (slope 1 when coming from the bottom)with Euclid length

p2, or, in other words, we do not change its moving at the wall, and

we see that the path of the particle in the region between x = 2m+ 1 and x = 4m is justthe reverse of the path of the particle when it goes from the right to the left for the firsttime. The algorithm can be similarly extended at the left wall to make the unfolded pathlook like the graph of a function of a single variable. It can be verified that when it passesthe point with the x-coordinate being a multiple of 2m, the movement stops. It can be

90 TYPE In2

seen that before it stops, the path in [2tm+1, 2(t+1)m] is just a copy of a particle pathin the above box of :

the t+22

th path from the left wall to the right wall if t is even

or t+12

th path from the right wall to the the left wall if t is odd.Therefore the above trick is just that we draw the picture on folded paper, then we

unfold this and see a simple picture in which the original problem can be tackled. Hereis an example for m= 5, k = 2 in Figure 5.3 being the unfolded case of Figure 5.2.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 19 2017 18

72

3

2

1

−12

Figure 5.3: the unfolded path for m= 5, k = 2

Lemma 5.2.1. Let l = lcm(k, m). The particle stops in the unfolded path when it moves2l − 1 for its x-coordinate. Furthermore(i) when l

mis even, the particle stops at (1,0);

(ii) when lm

and lk

are odd, the particle stops at (2m, 0);(iii) when l

mis odd and l

kis even, the particle stops at (2m, 2k− 1).

Proof. By prolonging the path at the beginning and the ending, respectively, by half unitfor x-coordinate to complete a period, we can consider the particle starting from the topand stopping at the top or the bottom. By observing the unfolded path of the particle,each time it goes from the top ceiling to the bottom ground or from the bottom to thetop, the x-coordinate is increased by 2k, so the first conclusion follows naturally. Theother two conclusions hold easily by basic number theory knowledge about congruence.Furthermore, elementary number theory tells us that both l

mand l

kcan not both be even,

which implies that the particle never stops at (1, 2k− 1).

Recall the definition of Brauer algebras of simply laced type, their Brauer monoidactions on admissible root sets and the classical diagram representation of Brauer algebraof type A in Chapter 1.

Lemma 5.2.2. The map φ defined on the generators in Theorem 5.1.3 induces a Z[δ±1]-algebra homomorphism.

Proof. We deal first with n is odd. In Definition 5.1.2, the nontrivial relations to beverified are just (5.1.18), (5.1.19), and (5.1.20). By [33], the relation (5.1.20) holds forthe inspection of the images of generators. The relation (5.1.19) follows by observing thediagrams in Br(A2m−2) because the top and the bottom are fixed, both of which have just

5.2 AN INTERESTING ELEMENTARY PROBLEM 91

one point having a vertical strand. By diagram we see that the action ofφ([r1r0r1 . . .]s) inW (A2m−2) (which we identify with Sym2m−1) on a number 1≤ a ≤ 2m−1, for s ≤ 2m−2,is given by

φ([r1r0r1 . . .]s)(a) =

a+ s, a odd, a+ s ≤ 2m− 1

a− s, a even, a− s ≥ 1

4m− 1− a− s, a odd, a+ s > 2m− 1

1+ s− a, a even, a− s < 1

(5.2.1)

Then for t > 0,φ([r1r0r1 . . .]2m−2)(2t) = 2m− 1− 2t,

φ([r1r0r1 . . .]2m−2)(2t + 1) = 2m− 1− 2t + 1.

Thereforeφ([r1r0r1 . . .]−1

2m−2)α2t = α2m−1−2t ,

φ([r1r0r1 . . .]−12m−2){α2t}m−1

t=1 = {α2t−1}m−1t=1 ,

which implies that φ([r1r0r1 . . .]−12m−2)φ(e0)φ([r1r0r1 . . .]2m−2) = φ(e1) and (5.1.18)

holds for the generator images under φ.Now consider φ when n = 2m > 5 even. The fact that (5.1.1)–(5.1.4), and (5.1.8)–

(5.1.11) still hold for the generator images under φ can be proved easily by Brauerdiagrams. As above we see that

φ([r1r0r1 . . .]s)(a) =

a+ s, a odd, a+ s ≤ 2m− 1

a− s, a even, a− s ≥ 1

4m+ 1− a− s, a odd, a+ s > 2m− 1

1+ s− a, a even, a− s < 1,

(5.2.2)

φ([r0r1r0 . . .]s)(a) =

a+ s, a even, a+ s ≤ 2m− 1

a− s, a odd, a− s ≥ 1

4m+ 1− a− s, a even, a+ s > 2m− 1

1+ s− a, a odd, a− s < 1.

(5.2.3)

By diagram inspection, we see that

φ([r1r0r1 . . .]−12m−1)(α2t) = α2m−2t ,

φ([r1r0r1 . . .]−12m−1)({α2t}m−1

t=1 ) = {α2t}m−1t=1 ,

hence φ([r1r0r1 . . .]2m−1)φ(e0)φ([r1r0r1 . . .]2m−1) = φ(e0); therefore relation (5.1.5)holds for the images of the generators under φ. On the other hand,

φ([r0r1r0 . . .]−12m−1)(α2t−1) = α2m−2t+1,

φ([r0r1r0 . . .]−12m−1)({α2t−1}mt=1) = {α2t}m−1

t=1 .

Just observing Brauer diagrams, (5.1.6) and (5.1.7) hold for the images of the generatorsunder φ.

92 TYPE In2

As for (5.1.12)–(5.1.14), consider the top and the bottom of the diagram of the leftunder φ; both the top and the bottom have horizontal the same strands among thosepoints {(i, 0)}2m−1

i=2 ∪ {(i, 1)}2m−1i=2 as φ(e0). Except those 2m− 2 strands in the top and

in the bottom fixed for φ(e0), there are still two strands of the left side under φ un-known. Those two strands are between the remaining four points, (0, 1), (0, 2m), (1, 1)and (1, 2m). If we find another end of the strand from (1, 1), the other strand is fixed asa consequence. The strands starting from (1,1) in the images of under φ the right handsides of (5.1.12)–(5.1.14) are ended at (0, 2m), (0,1) and (1, 2m), respectively. By obser-vation, we can transform this equality problem to the elementary problem solved at thebeginning of this section in the following way. Consider the paths of a particle startingfrom (1, 1) in the diagram of the left hand sides of the images under φ of (5.1.12)–(5.1.14) with the m−1 horizontal strands at the top and the m−1 horizontal strands atthe bottom removed and transform the horizontal strands as in Figure 5.4. We give anexample for this in Figure 5.5 for φ(e0)φ(r1)φ(r0)φ(r1)φ(e0) in Br(A7). By observation,Lemma 5.2.1 can be applied here, and gives that the three equalities hold under φ actingon both sides.

−→

−→

Figure 5.4: transformation of horizontal strands

Figure 5.5: φ(e0)φ(r1)φ(r0)φ(r1)φ(e0) for I82

5.3 NORMAL FORMS FOR BrM(In2) 93

5.3 Normal forms for BrM(In2)

Lemma 5.3.1. The submonoid generated by r0 and r1 in BrM(In2) is isomorphic to W (In

2).

Proof. The lemma follows from the natural homomorphisms chain below.

Z[δ±1](W (In2))→ Br(In

2)→ Br(In2)/(e0, e1)→ Z[δ±1](W (In

2)).

The composition is the identity and so the lemma follows.

From now on, we do not distinguish between W (In2) in BrM(In

2) and its image underφ.As in Remark 1.5.6, we can also define an anti-involution on Br(In

2) denoted by x 7→ xop.

Proposition 5.3.2. The natural anti-involution above induces an automorphism of theZ[δ±1]-algebra Br(In

2).

Proof. It suffices to check the defining relations given in Definition 5.1.1 and Definition5.1.2 still hold under the anti-involution. An easy inspection shows that all relationsinvolved in the definition are invariant under op, except for (5.1.6), (5.1.7), (5.1.8),(5.1.9), and (5.1.12). The relation obtained by applying the anti-involution to (5.1.12)holds due to (5.1.5). The equalities (5.1.6) and (5.1.8) are the op-duals of (5.1.7) and(5.1.9), respectively. Hence our claim holds.

Proposition 5.3.3. Suppose that D2m−10 is a set of left coset representatives of the subgroup

generated by r0 in W (I2m−12 ). Up to some powers of δ, each element in BrM(I2m−1

2 ) can bewritten as an element in W (I2m−1

2 ) or as ue0v, where u ∈ D2m−10 and v ∈ (D2m−1

0 )op.

Proof. By (5.1.18), it follows that e1 is conjugate to e0 under W (I2m−12 ); hence we only

need to prove that D2m−10 e0(D

2m−10 )op is closed under multiplication by the generators

e0, r0 and r1 up to some power of δ. By Proposition 5.3.2, and invariance of the setD2m−1

0 e0(D2m−10 )op under the natural involution, it suffices to prove it is closed under left

multiplication. For r0 and r1, we just apply (5.1.16). For e0, it follows from (5.1.19).

Let Ψ be the root system of In2, and Ψ+ be the positive roots with respect to β0, β1

which are roots corresponding to r0, r1, respectively. We consider the natural action ofW (In

2) on Ψ+ by negating the negative roots once these appears in action.

Lemma 5.3.4. Let N0, N1 be the stabilizers of β0 and β1 in W (I2m2 ), respectively. Then for

any element a ∈ Ni , we have aeia−1 = ei , for i = 0, 1.

Proof. When n is odd, Ni =

ri�

for i = 0, 1. Hence the lemma holds because

riei ri(5.1.16)= ei .

When n= 2m even, we have that N0 =

r0, [r1r0 · · · ]2m−1�

and N1 =

r1, [r0r1 · · · ]2m−1�

;hence the lemma holds because of the following equalities.

riei ri(5.1.2)= ei ,

�

r1r0 · · · ]2m−1e0[r1r0 · · ·�

2m−1(5.1.5)= e0,

�

r0r1 · · · ]2m−1e1[r0r1 · · ·�

2m−1(5.1.6)+(5.1.7)= e1.

94 TYPE In2

Consider a positive root β and a node i of type In2. If there exists w ∈ W such that

wβi = β , then we can define the element eβ in BrM(In2) as (1.6.30) by

eβ = weiw−1.

The above lemma implies that eβ is well defined.

Lemma 5.3.5. Let D2mi be a set of left coset representatives for Ni in W (I2m

2 ) for i = 0, 1,and K0 = ⟨[r1r0 · · · ]2m−1⟩ ⊂ N0, K1 = ⟨1⟩ ⊂ N1. Then, for any r ∈ W (I2m

2 ), there exista ∈ Di and b ∈ Ki , such that

rei = aei b.

Proof. It is a direct consequence of (5.1.2), (5.1.5) and (5.1.7).

Proposition 5.3.6. Up to some power of δ, each element in BrM(I2m2 ) can be written as

(i) a ∈W (I2m2 ),

(ii) uei vw , u ∈ D2mi , v ∈ Ki , w ∈ (D2m

i )op for i = 0, 1,

(iii) u′e0e1w′, u′ ∈ D2m0 , w′ ∈ (D2m

1 )op,

(iv) u′′e1e0w′′, u′′ ∈ D2m1 , w′′ ∈ (D2m

0 )op,

(v) u′′′e0e1e0w′′′,u′′′ ∈ D2m0 , w′′′ ∈ (D2m

0 )op.

Proof. Let us first prove the claim that the monomial e0re1 can be written as e0e1 forany r ∈ W (I2m

2 ) up to some power of δ. In view of (5.1.2), we only need consider theelements that can be written as [r1r0 · · · ]2k with 2k ≤ 2m. Also thanks to (5.1.8), we canrestrict ourselves to m< 2k ≤ 2m, follows from below,

e0[r1r0 · · · ]2ke1(5.1.1)= e0[r0r1 · · · ]2m−2k−1[r0r1 · · · ]2m−1e1

(5.1.7)= e0r0[r1r0 · · · ]2m−2k−2e1

(5.1.2)= e0[r1r0 · · · ]2m−2k−2e1

(5.1.8)= δθm−k−1 e0e1.

It follows that for any r ∈ W (I2m2 ), the monomial ei re j can be written as one of e0,

[r1r0 · · · ]2m−1e0 e1, e0e1, e1e0, and e0e1e0 up to some power of δ.To prove the lemma, it remains to prove that those five kinds of normal forms are

closed under multiplication by generators ei . Thanks to Proposition 5.3.2, we only needconsider multiplication from the left. By the conclusion from the above paragraph, thelemma holds.

5.4 THE RANK OF Imφ 95

5.4 The rank of Imφ

For establishing Theorem 5.1.3, it suffices to prove that the normal forms in Propositions5.3.3 and 5.3.6 represent different diagrams in Br(An−1). The problem can be reduced tothe counting of orbit sizes.(I) If n= 2m− 1 odd, then

#φ(W (I2m−12 ))({α2t}m−1

t=1 ) = 2m− 1,

(II) If n= 2m even, then

#φ(W (I2m2 ))({α2t}m−1

t=1 ) = m,

#φ(W (I2m2 ))({α2t−1}mt=1) = m,

#φ(W (I2m2 ))(φ(e0){α2t−1}mt=1) = m,

and the last two orbits are different.First we consider n= 2m− 1≥ 5.By (5.2.1), when 0< s < 2m−1, we see that s+1 is not occupied in the horizontal strandsof φ([r1r0r1 · · · ]−1

s )({α2t}m−1t=1 ). Then #φ(W (I2m−1

2 ))({α2t}m−1t=1 ) is at least 2m− 1. But

the subgroup

r0�

stabilizes {α2t}m−1t=1 ; therefore by Lagrange’s Theorem, (I) holds.

When n= 2m> 5, we define

α = Σ2m−1i=1 αi ,

Y0 = {α2t}m−1t=1 ,

Y1 = {α2t−1}mt=1,

Y2 = φ(e0)Y1 = Y0 ∪ {α}.

Now we consider the case when m = 2m′ + 1 is odd. Here we denote by h(γ) theheight of γ ∈ Φ+ which means the sum of the coefficients of simple roots for γ writtenas the linear combination of simple roots (see Definition 1.1.5). We find that when0≤ s ≤ m− 2,

max{h(γ) | γ ∈ φ([r1r0 · · · ]−1s )(Y0)}= h(φ([r1r0 · · · ]−1

s )(α2m′)) = 2s+ 1,

max{h(γ) | γ ∈ φ([r1r0 · · · ]−12m′)(Y0)}= h(φ([r1r0 · · · ]−1

2m′)(α2m′)) = 4m′.

Then it follows that #W (I4m′+22 )(Y0) is at least m. At the same time

r0, [r1r0 · · · ]2m−1�

stabilizes Y0; therefore it follows from Lagrange’s Theorem that #W (I4m′+22 )(Y0) is exactly

m.Similarly we see that when 0≤ s ≤ m− 1

max{h(γ) | γ ∈ φ([r0r1 · · · ]−1s )(Y1)}= h(φ([r0r1 · · · ]−1

s )(αm)) = 2s+ 1.

Thus W (I4m′+22 )(Y1) has at least m elements. At the same time

r1, [r0r1 · · · ]2m−1�

stabi-lizes Y1, and so by Lagrange’s Theorem, #W (I4m′+2

2 )(Y1) = m.When we consider W (I4m′+2

2 )(Y2), we need some result from Chapter 2. By Section 2.4,

96 TYPE In2

we see that Y2 consists of m′ symmetric pairs and 1 symmetric roots, and this numericalinformation is not changed under W (I4m′+2

2 ) ⊂ W (C2m′+1) (Weyl group of type C2m′+1

in Chapter 2). The orbit W (I4m′+22 )({α}) has at least m elements, hence using the same

argument as the above, we see that #W (I4m′+22 )(Y2) is also exactly m.

To prove that the orbits of Y1 and Y2 have no intersection, it suffices to verify that Y2 isnot in the orbit of Y1. By the above, we see that α only occurs in Y = φ([r0r1 · · · ]2m′)(Y1)in the orbit of Y1 under W (I2m

2 ). But h(φ([r0r1 · · · ]2m′)(αm−2)) = 4m′ − 2 > 1, whichcontradicts the heights of elements in Y \ {α}. With (II) verified, we have proved theTheorem 5.1.3 for n≡ 2 mod 4, and n≥ 5.At last, consider the case when n = 2m ≥ 5, and m = 2m′. The formula #W (I4m′

2 )(Y1) =#W (I4m′

2 )(Y0) = m can be proved by the same argument as the above.From Section 2.4, we see that Y1 has m′ pairs of symmetric roots and no symmetric root,and Y2 has m′−1 pairs and 2 symmetric roots α and α2m′ . Hence the W (I2m

2 )-orbits of Y1and Y2 have no intersection.When 0≤ s < 2m′ − 3, we have

max{h(γ) | γ ∈ φ([r1r0 · · · ]−1s+1)(Y2 \ {α2m′ ,α})}

>max{h(γ) | γ ∈ φ([r1r0 · · · ]−1s )(Y2 \ {α2m′ ,α})},

so the orbit of Y2 has at least 2m′−2 elements. Therefore the cardinality of the stabi-lizer in W (Im

2 ) is smaller than 8m′

2m′−2. If m′ > 3, 8m′

2m′−2< 6, but the group

r0, [r1r0 · · · ]4m′−1�

stabilizes Y2, hence the subgroup will be the full stabilizer. By checking when m′ = 2,3, finally, we see that #W (I4m′

2 )(Y1) = 2m′ = m. With (II) verified, we have proved themain theorem for n≡ 0 mod 4, and n≥ 5.

Now we have the following decomposition of Br(In2) as a Z[δ±1]-module.

Br(In2) = Br(In

2)/(e0)⊕ (e0), 2 - n,

Br(In2) = Br(In

2)/(e0)⊕ (e0)/(e0e1e0)⊕ (e0e1e0), 2 | n.

Therefore the theorem below about the cellularity can be obtained by an argument simi-lar to [2] and Section 4.6.

Theorem 5.4.1. If R is a field such that the group ring R[W (In2)] is a cellular algebra, then

the algebra Br(In2)⊗ R is a cellularly stratified algebra.

Remark 5.4.2. For the hypothesis of the Theorem 5.4.1, with the method in [24], weconjecture that a sufficient condition for R[W (In

2)] is a cellular algebra is that the charac-teristic of R does not divide n.

6TYPES H3 AND TYPE H4

6.1 Introduction

In this chapter, we present Brauer algebras of type H3 and H4 associated to sphericalCoxeter groups of type H3 and H4 by using the idea of admissible partition of Mühlherr([33]), also used in Chapter 5. Their definitions are given in Definition 6.2.1. Thosetwo algebras can be considered as subalgebras of Br(D6) and Br(E8), respectively ([9] or[17]). The diagrams H3, H4, D6 and E8 are presented in Figure 6.1; here the diagramsof D6 and E8 are specially depicted for Mühlherr’s admissible partition corresponding tothe diagram of H3 and H4 in such a way that the parts are the sets of nodes on a verticalline. Definitions are given in Section 6.2.

In this chapter, we will present the following two main theorems about Br(H3) andBr(H4), respectively. To avoid confusion, the generators of Br(D6) and Br(E8) are capital-ized.

Theorem 6.1.1. There exists an injective Z[δ±1]-algebra homomorphism

φ1 : Br(H3)−→ Br(D6)

determined by φ1(r1) = R2R4, φ1(r2) = R3R5, φ1(r3) = R1R6, φ1(e1) = E2E4, φ1(e2) =E3E5 and φ1(e3) = E1E6. Furthermore Br(H3) is free of rank 1045 over Z[δ±1].

1

2

3

4

5 6

13

2

4

5

7 8

1 2 3 1 2 3 4

5 5H3 H4

D6 E8

6

Figure 6.1: Coxeter diagrams of H3, H4, D6 and E8

98 TYPES H3 AND TYPE H4

Theorem 6.1.2. There exists an injective Z[δ±1]-algebra homomorphism

φ2 : Br(H4)−→ Br(E8)

determined by φ2(r1) = R2R5, φ2(r2) = R4R6, φ2(r3) = R3R7, φ2(r4) = R1R8, φ2(e1) =E2E5, φ2(e2) = E4E6, φ2(e3) = E3E7 and φ2(e4) = E1E8. Furthermore Br(H4) is free ofrank 236025 over Z[δ±1].

6.2 Definition

Let δ be a generator of an infinite cyclic group.

Definition 6.2.1. For k = 3, 4, the Brauer algebra of type Hk, denoted by Br(Hk), is aunital associative Z[δ±1]-algebra generated by {ri , ei}ki=1, subject to (1.6.1)–(1.6.9) withκi = 2, ξ j = 0 and the following relations.

r1r2r1r2r1 = r2r1r2r1r1, (6.2.1)

r1r2e1r2r1 = r2r1e2r1r1, (6.2.2)

e1e2e1 = e1, (6.2.3)

e1r2e1 = e1, (6.2.4)

e1r2r1r2e1 = e1, (6.2.5)

and the additional relation for k = 4

e1(r2r1r2r1r3r2r1r2r3r1r2r1r2r3r4)5 = e1. (6.2.6)

The submonoid of the multiplicative monoid of Br(Hk) generated by δ, {ri , ei}ki=1 is de-noted by BrM(Hk). This is the monoid of monomials in Br(Hk).

If we just focus on relations with {r1, r2, e1, e2} involved only, we find a homomorphicimage of the algebra Br(I5

2) in Chapter 5 with the given parameters κi = 2 and ξ j = 0as in Definition 6.2.1, which is also isomorphic to the algebra BI5

(γ) in [8] up to someparameters.

6.3 Admissible root sets

Let {βi}4i=1 be simple roots of W (H4) corresponding to the notation of Figure 6.1 ({βi}3i=1for W (H3)). They can be embedded into Euclidean space R4 with each of them havingEuclidean length

p2 and (β1,β2) =

−p

5−12=−ϕ with ϕ2 = ϕ+1. Let Ψ4 and Ψ3 denote

the root systems of W (H4) and W (H3), respectively, and Ψ+4 and Ψ+3 the positive rootsrespecting {βi}4i=1. It is known from [1] that #Ψ+3 = 15 and #Ψ+4 = 60. A mutuallyorthogonal subset B ⊂ Ψ+4 (Ψ+3 ) is called an orthogonal basis if B spans R4 (R3). It canbe verified from the reflections in W (H4) (W (H3)) by GAP ([23]) that any two mutuallyorthogonal subsets (commutative reflections) of the same cardinality are in the sameorbit under W (H4) (W (H3)). Here we just consider the natural Coxeter group action

6.3 ADMISSIBLE ROOT SETS 99

W (H4) (W (H3)) on Ψ4 (Ψ3) acting on the positive roots by negating the negative ones.This corresponds to the action by conjugation on reflections.Let

β5 = ϕ2β1 + 2ϕβ2 +ϕβ3 = r2r1r2r1r3r2β1 ∈Ψ3,

and let r5 be the reflection with root β5 in W (H3) and W (H4). Let N31 and N4

1 be thestabilizers of β1 in W (H3) and W (H4), respectively.

Lemma 6.3.1. We have that

N31 =

r1, r3, r5�∼=W (A1)

3

N41 =

r1, r3, r4, r5�∼=W (A1)×W (H3).

Proof. We have the following results of inner products involving β5,(β1,β5) = 0= (β3,β5), (β4,β5) =−ϕ.Hence we have the diagram relation for them as below which indicates that our claimabout the group isomorphisms in this lemma holds. Using this observation, we can com-pute indices of the two subgroups

r1, r3, r5�

and

r1, r3, r4, r5�

in W (H3) and W (H4)and show that these are 15= #Ψ+3 and 60= #Ψ+4 , respectively.

◦1 ◦3 ◦45 ◦5

By the diagram, we find that the two (resp. three) reflections stabilize β1 in W (H3) (resp.W (H4)), respectively. Therefore the lemma follows from Lagrange’s Theorem.

It can be verified that the subgroup

(r5r3r4)5�

is a normal subgroup of

r3, r4, r5�∼=

W (H3) and has cardinality 2 because (r5r3r4)10 = 1. Let C31 =

r3, r5�

and C41 be the quo-

tient group of

(r5r3r4)5�

in

r3, r4, r5�

. Then C31∼=W (A1)2 and C4

1∼=W (H3)/

(r5r3r4)5�

.Let D3

1 and D41 be sets of left coset representatives of of N3

1 in W (H3) and N41 in W (H4),

respectively. Then #D31 = #Ψ+3 = 15, #D4

1 = #Ψ+4 = 60. As Lemma 1.6.6, we have thefollowing lemma.

Lemma 6.3.2. For i and j be nodes of the Coxeter diagram of H3 or H4. If w ∈W (H4) orW (H3), satisfies wβi = β j , then weiw

−1 = e j .

Proof. It suffices to prove that any generator of N41 in Lemma 6.3.1, satisfies that re1r−1 =

e1. The cases of r1, r3, r4 hold trivially; but for r5, we just apply the following formulaswhich can be deduced easily from the definition,

r2r1r2r1e2r1r2r1r2 = e1,

r1r2r1r2e1r2r1r2r1 = e2,

r2r3e2r3r2 = e3,

r3r2e3r2r3 = e2,

then we have

r5e1r5 = r2r1r2r1r3r2r1r2r3(r1r2r1r2e1r2r1r2r1)r3r2r1r2r3r1r2r1r2

= r2r1r2r1r3r2r1(r2r3e2r3r2)r1r2r3r1r2r1r2

= r2r1r2r1r3r2(r1e3r1)r2r3r1r2r1r2

= r2r1r2r1(r3r2e3r2r3)r1r2r1r2

= r2r1r2r1e2r1r2r1r2 = e1.


We define eβ = rei r−1, if β = rβi . By the above lemma it is well defined. Since

Coxeter groups W (H3) and W (H4) acts transitively on mutually orthogonal root sets ofthe same cardinality, the equality e1e3 = e3e1 implies that eβ eβ ′ = eβ ′ eβ , if {β ,β ′} is amutually orthogonal root set. Therefore, for any mutually orthogonal root subset B ofΨ+3 or Ψ+4 , as (1.6.30) we can define that

eB =∏

β∈B

eβ .

Lemma 6.3.3. For Ψ+3 and Ψ+4 , the following holds.

(i) For each β ∈Ψ+3 , there is a unique orthogonal basis subset of Ψ+3 containing β; hencethere are 5 different orthogonal basis subsets of Ψ+3 .

(ii) For each β ∈Ψ+4 , there are 5 orthogonal basis subsets of Ψ+4 containing it; hence thereare 75 different orthogonal basis subsets of Ψ+4 .

(iii) For any orthogonal subset B of Ψ+4 with #B > 1, there is a unique orthogonal basissubset of Ψ+4 containing B.

Proof. Any β can be obtained from β1 by acting some element in W (H3), so we only needconsider β = β1. If β1 ∈ B ⊂ Ψ+3 is an orthogonal basis, then there exist another twomutually orthogonal positive roots β ′, β ′′ in B that are orthogonal to β1, which impliesthat the corresponding reflections can fix β1. By Lemma 6.3.1, we see that B must be{β1,β3,β5}. Hence the number of orthogonal basis subsets of Ψ+3 is 15

3= 5.

The first conclusion of (ii) follows from (i) and Lemma 6.3.1. The second one holds as60×5

4= 75.

Assertion (iii) follows from (ii) and (i).

Definition 6.3.4. If a mutually orthogonal subset B of Ψ+3 (Ψ+4 ) has at most one elementor is an orthogonal basis, then we call B an admissible root set of type H3 (H4) or Badmissible.

Remark 6.3.5. By Lemma 6.3.1 and Lemma 6.3.3, there exists a unique positive root forC4

1∼= W (H3) with β3, β4, and β5 being the simple roots of C4

1 that are orthogonal toβ1. There exists a unique positive root orthogonal to {β1,β3,β5} and we denote it by β7.Hence {β1,β3,β5,β7} is an orthogonal basis and admissible root set of type H4.

By Lemma 6.3.3, for each mutually orthogonal subset B of Ψ+3 (Ψ+4 ), there exists a

minimal unique admissible root set containing B; we denote it by Bcl, and call it theadmissible closure of B. As Lemma 3.3.6, the lemma below holds.

Lemma 6.3.6. For each mutually orthogonal subset B of Ψ+3 (Ψ+4 ), we have

eBcl = δ2#(Bcl\B)eB.

Let N32 and N4

2 be stabilizers of {β1,β3,β5} in W (H3) and {β1,β3,β5, β7} in W (H4),respectively. Let D3

2 and D42 be sets of coset representatives of N3

2 in W (H3) and N42 in

W (H4), respectively. Then #D32 = 5, #D4

2 = 75 by Lemma 6.3.3.

6.4 NORMAL FORMS FOR BrM(H3) AND BrM(H4) 101

6.4 Normal forms for BrM(H3) and BrM(H4)

As Proposition 3.4.7, the following conclusion can be verified.

Lemma 6.4.1. In BrM(H3) (BrM(H4)), the following holds.

(i) Each element in C31 (C

41 ) commutes with e1.

(ii) For each element r ∈ W (H3) (W (H4)), there exist r ′ ∈ D31 (D

41) and r ′′ ∈ C3

1 (C41 ),

such thatre1 = r ′e1r ′′.

(iii) For each element r ∈W (H3) (W (H4)), there exists an element r ′ ∈ D32 (D

42), such that

re1e3 = r ′e1e3.

Next we consider the cases for Temperley-Lieb elements.

Lemma 6.4.2. Let β ∈ Ψ+3 (Ψ+4 ) such that β is not equal to or orthogonal to β1. Then

there exists some r ′ ∈ D31 (D

41) and r ′′ ∈ C3

1 (C41 ), such that

eβ e1 = r ′e1r ′′.

Proof. If (β1,β) = ±1, since there exists such an element r in W (H3) or W (H4) thatr{β1,β} = {β2,β3}; then eβ e1 = r1rβ e1 in view of (1.6.8). Consequently, the lemmafollows from Lemma 6.4.1. If (β1,β) 6= ±1, then β can be obtained by letting someelement from N3

1 or N41 act on some root of the linear combination of β1 and β2; hence

it can be reduced to the case of BrM(I52) which has been verified in Proposition 5.3.3.

Therefore the lemma also holds under this condition.

By Proposition 5.3.3, the corollary below can be shown to hold.

Corollary 6.4.3. Let β ∈ Ψ+3 (Ψ+4 ). Then up to some power of δ, there exist some r ′ ∈ D3

2(D4

2), and k ∈ Z, such thateβ e1e3 = δ

k r ′e1e3.

Proof. We consider the corollary for Br(H4), and the conclusion for Br(H3) can be provedsimilarly. By Lemma 6.3.6, we know that e1e3eβ5

eβ7= δ4e1e3. No β ∈ Ψ+4 can be

orthogonal to {β1,β3,β5,β7}. If β ∈ {β1,β3,β5,β7}, then eβ e1e3 = δ2e1e3. If β is notin {β1,β3,β5,β7}, there exists one βi , i ∈ {1,3, 5,7}, such that

¬

rβ , rβi

¶

is isomorphic toW (I5

2) or W (A2); under the conjugation of W (H4), the corollary follows from Proposition5.3.3 and (1.6.8).

Theorem 6.4.4. Up to some power of δ, for each element in BrM(Hk) for k = 3, 4, it canbe written in one of the three normal forms below.

(i) r ∈W (Hk),

(ii) ue1vw, u ∈ Dk1, w−1 ∈ Dk

1, v ∈ C k1 ,

(iii) ue1e3w, u ∈ Dk2, w−1 ∈ Dk

2.


Proof. As in Remark 1.5.6, we can define a natural involution on Br(Hk), by reversing theeach monomial in BrM(Hk), denoted by ·op. It can be verified that ·op induces a naturalisomorphism on Br(Hk) and the difficult one is (6.2.6). By Lemma 6.3.2 and Definition6.2.1, we know that e1 commutes with r3, r4, r5. Since (r5r3r4)5 has order 2 we havethat ((r5r3r4)5)op = (r5r3r4)5. Then the left side of (6.2.6) can be written as

e1(r5r3r4)5 = (r5r3r4)

5e1 = (r5r3r4)5)ope1 = (e1(r5r3r4)

5)op.

Consequently, the equality (6.2.6) still holds after application of ·op.Note that 1 is a normal form. To prove the theorem, it suffices to prove that the above

forms are closed under left multiplication by ri for i = 1, . . ., k and by eβ for β ∈ Ψ+k .Hence the lemma follows from Lemmas 6.3.6, 6.4.1, 6.4.2 and Corollary 6.4.3.

Remark 6.4.5. The numbers of the normal forms in Theorem 6.4.4 are

120+ 4× 152 + 52 = 1045,

14400+ 60× 602 + 752 = 236025

for Br(H3) and Br(H4), respectively.

6.5 Images of φ1 and φ2

Let {αi}6i=1 be the simple roots of W (D6) (Weyl group of type D6) corresponding to thediagram of D6 in Figure 6.1, and let Φ+6 be the positive roots of W (D6). From [9, Propo-sition 4.9, Proposition 4.1], up to some power of δ, there is a normal form associatedto admissible root sets of type D6, which are the orbits of B3

0 = ;, {α2}, B31 = {α2,α4},

{α2,α4,α6} and B32 = {α1,α2,α4,α∗4,α6,α∗6} for each element of BrM(D6), where α∗ is

still the orthogonal mate for each positive root α of type D6 as in Section 3.3. Now wecan prove the injectivity of φ1 and Theorem 6.1.1 by analyzing the image of each formin Theorem 6.4.4.

Proof. By checking the homomorphism of Br(I52) into Br(A4) of Theorem 5.1.3, it is easy

to verify that φ1 is an algebra homomorphism.By [33], the normal forms in (i) of Theorem 6.4.4 are embedded into BrM(D6) by φ1.It can be verified that φ1(r5) = R∗4R∗6, where R∗4 and R∗6 are reflections correspond-ing to α∗4 and α∗6. By the diagram representations for Br(D6) in [13] and Proposition1.3.3, φ1(C3

1 ) =¬

R1R6, R∗4R∗6¶

is embedded into W (CB31) (Proposition 1.3.3). Since

φ1(e1) = E2E4 and φ1(W (H3))B31 has 15 different subsets of Φ+6 , each normal form in

(ii) of Theorem 6.4.4 has a unique image of the normal forms for BrM(D6) in Proposition1.3.3 with X ∈W (D6)(B3

1).The same conclusion for normal forms in (iii) of Theorem 6.4.4 is embedded for X ∈W (D6)(B3

2), because φ1(e1e3eβ5) = EB3

2, and φ1(W (H3))B3

2 has 5 different subsets of Ψ+6 .Therefore φ1 is an injective homomorphism.

Considering Table 1.2, we give the version of Theorem 1.5.11 for type E8. Let

Y ∈ Y = {;, {2}, {2, 5}, {2, 3,5, 7}}

6.5 IMAGES OF φ1 AND φ2 103

for type E8. Each monomial a in BrM(E8) can be uniquely written as δiaB eY haopB′ for some

i ∈ Z and h ∈W (MY ) in 1.2, where B = a;, B′= ;a, aB ∈ BrM(E8), aop

B′ ∈ BrM(E8) and(i) a;= aB;= aBBY , ;a = ;aop

B′ = BY aopB′ ,


In view of E5E6E7E2E4E5{α6,α8}= {α2,α5} and ht(E5E6E7E2E4E5) = 0, we have

W (M{2,5}) = E5E6E7E2E4E5W (M{6,8})E5E4E2E7E6E5∼=W (A5)

=

E2 E5R3, E2 E5R1, E2 E5R8E6E4E5E7R3E4E6E5E2, E2 E5R8, E2 E5R7�

,

W (M{2,3,5,7}) = {1},

where Ei = δ−1Ei . Let R′1 = E2 E5R3, R′2 = E2 E5R1, R′3 = E2 E5R8E6E4E5E7R3E4E6E5E2,R′4 = E2 E5R8, R′5 = E2 E5R7. These monomials can be considered the natural generatorsof W (A5) with their indices corresponding to the original Dynkin diagram of type A5 byuse of [17, Proposition 4.3]. Let α′ = 2α4 + α2 + α3 + α5 and α′′ = α2 + α3 + 2α4 +2α5 + 2α6 + α7. In view of the natural embedding of W (H4) into W (E8) through φ2, itcan be checked that φ2(r5) = Rα′Rα′′ , and φ2(e1r5) = E2E5Rα′Rα′′ = δ2R′1R′3. After thesepreparations, we prove Theorem 6.1.2 in the below.

Proof. As in proving φ1 is an algebra homomorphism, it can be verified that (1.6.1)-(1.6.9) and (6.2.1)–(6.2.5) still hold under φ2. Each element in {α1,α3,α′,α′′,α7,α8} isorthogonal to {α2,α5}, so by definition of R′i , we have

φ(e1(r5r3r4)5) = E2E5(Rα′Rα′′R3R7R1R8)

5

= δ2(E2 E5Rα′ E2 E5Rα′′ E2 E5R3 E2 E5R7 E2 E5R1 E2 E5R8)5

= δ2(R′1R′3R′1R′5R′2R′4)5

= δ2(R′3R′5R′2R′4)5.

Applying the Mühlherr’s partition for W (I52) in W (A4) (with generators {R′i}

5i=2), the

above is equal to δ21W (M{2,5}) = δ2 E2 E5 = E2E5 = φ2(e1), therefore Equality (6.2.6)

holds under φ2.The remaining of the proof is dedicated to proving the injectivity of φ2. By [33], thenormal forms in (i) of Theorem 6.4.4 is embedded into W (E8) or the rewritten forms inTheorem 1.5.11 associated to ; by φ2.It can be checked that φ2(W (H4)){α2,α5} ⊂ A has cardinality 60. By the above wesee that there is a group homomorphism C4

1 → W (M{2,5}) defined by x → δ−2φ2(e1 x),with the image being the subgroup of cardinality 60 generated by {R′1R′3, R′2R′4, R′1R′5}W (M{2,5}). Since #C4

1 = 60, it is an isomorphism. The normal forms in (ii) of Theorem6.4.4 are embedded into the normal forms in Theorem 1.5.11 associated to {2,5} by φ2.It can be also verified that the subset φ2(W (H4)){α2,α5,α3,α7}cl ⊂ A has cardinality75. Consequently, the normal forms in (iii) of Theorem 6.4.4 are embedded into thenormal forms in Theorem 1.5.11 associated to {2,3, 5,7} by φ2.

Remark 6.5.1. The isomorphism from C41∼= W (H3)/

(r3r5r4)5�

to W (M{2,5}) ∼= W (A2),can be considered induced from the composition of the canonical homomorphisms

W (H3)f−→W (D6)

g−→W (A5)

with

(r3r5r4)5�

being the kernel of g f .


Now we have decompositions for Br(H3) and Br(H4) as Z[δ±1]-module as follows.

Br(Hk) = Br(Hk)/(e1)⊕ (e1)/(e1e3)⊕ (e1e3), k = 3, 4.

Therefore the theorem below about the cellularity can be obtained by an argument simi-lar to [2] and Section 4.6.

Theorem 6.5.2. If R is a field such that the group ring R[W (H3)] (R[W (H4)]) is a cellularalgebra, then Br(H3)⊗ R (Br(H4)⊗ R) is a cellularly stratified algebra.

Remark 6.5.3. For the hypothesis of the Theorem 6.5.2, with the method in [24], we con-jecture that a sufficient condition is that the characteristic of R does not divide #W (H3)(#W (H4)).

Remark 6.5.4. Up to some coefficients, Br(H3) is isomorphic to BrGH3(γ) (Chen’s Brauer

algebra associated to W (H3) in [8]). Up to some coefficients, we can obtain BrGH4(γ) by

(1.6.1)-(1.6.9) and (6.2.1)–(6.2.5) in Definition 6.2.1, which can be proved to have rank452025 by modifying the C4

1 to

r3, r4, r5�

and the analogue argument in [8, Section 7]for BrGH3

(γ).

7TYPE G2

7.1 Introduction

We wrote about obtaining the Coxeter group of type G2 from the one of type D4 in Section1.6. In this chapter, we carry out the analogous operation on Br(D4) to obtain Br(G2) asa subalgebra of it. First recall the definition of Br(G2) in Definition 1.6.1. In Section1.6, we introduce that W (G2) can be obtained as a subgroup of from W (D4) as the fixedsubgroup of the isomorphism σ = (1, 2,4) acting on the generators of W (D4) indicatedin Figure 7.1.

Here the action of σ can be extended to an isomorphism onto the Brauer algebraof type D4 by acting on the Temperley-Lieb generators Ei ’s on their indices. We denoteSBr(D4) the subalgebra of Br(D4) generated by σ-invariant elements in BrM(D4). Themain theorem of this chapter can be stated as follows.

Theorem 7.1.1. There is an algebra isomorphism

φ : Br(G2)−→ SBr(D4)

defined by φ(r0) = R1R2R4, φ(r1) = R3, φ(e0) = E1E2E4 and φ(e1) = E3. Furthermore,both Br(G2) and SBr(D4) are free over Z[δ±1] of rank 39.

1

2

34

Figure 7.1: Dynkin diagram of type D4

106 TYPE G2

7.2 Root systems for D4 and G2

Let {αi}4i=1 be simple roots of D4. These can be realized in R4 with α1 = ε1 + ε2, αi =εi − εi−1 for 2 ≤ i ≤ 4, where {εi}4i=1 is an orthonormal basis of R4. The positive rootsystem will be taken to the vectors ε j ± εi with 1 ≤ i < j ≤ 4, denoted by Φ+, the wholeroot system is denoted by Φ. We can define a linear transformation by the followingmatrix, also called σ, by considering each element in R4 as a column vector.

−1/2 −1/2 −1/2 1/21/2 1/2 −1/2 1/21/2 −1/2 1/2 1/2−1/2 1/2 1/2 1/2

It can be checked that σ(α1) = α2, σ(α2) = α4, σ(α3) = α3, σ(α4) = α1 and σ has order3 in SL(R4).The Reynold’s map is a surjective linear map from R4 to the subspace of σ-invariantvectors, defined as follows,

R4 −→ R4σ

p : x 7→x +σ(x) +σ2(x)

3.

In particular,

β0 = p(α1) =α1 +α2 +α4

3, β1 = p(α3) = α3

is a basis of R4σ whose elements have square norm 2/3 and 2, respectively; these two

vectors can be taken as simple roots of G2. The root system of type G2 is denoted by Ψ,and the positive root system in which β0 and β1 are simple roots is denoted by Ψ+.

Recall the notion of admissible set and B from Definition 1.6.4. There is a naturalaction of the Coxeter group W (G2) (W (D4), respectively) acting onB (A , respectively),by negating roots in Ψ \Ψ+ (Φ \Φ+, respectively).

Proposition 7.2.1. The setB has two W (G2)-orbits, which are orbits of {β1} and {β0, 3β0+2β1}.

7.3 The map φ inducing a homomorphism

In order to avoid confusion with the above generators, the symbols of the generators ofBr(D4) have been capitalized.

Remark 7.3.1. In Section 1.3, we have seen that for each γ ∈ Φ and each orthogonalroot set X ⊂ Φ+, the monomials Eγ and EX are well defined. Let Rγ denote the reflectioncorresponding to γ. By conjugation of elements of W (D4), the symbol "∼" can have amore general meaning of two roots being not orthogonal or equal.

We prove that φ induces a homomorphism.

Proof. It suffices to verify that the relations involved in Definition 1.6.1 for type G2 stillhold when the generators are substituted by their images under φ.

7.4 NORMAL FORMS OF BrM(G2) 107

As for (1.6.18), (1.6.20)–(1.6.22), they follow from

φ(r0)φ(e1)φ(e0) = R1R2(R4E3E4)E1E2

(1.5.2)= R1(R2R3(E4E2))E1

(1.3.6)+(1.5.2)= R1E3E4E2E1

(1.3.6)+(1.5.2)= R3E4E2E1 = φ(r1)φ(e0),

φ(e1)φ(r0)φ(e1)φ(r0)φ(e1) = E3EΣ4i=1αi

E3(1.5.5)= E3 = φ(e1),

φ(e1)φ(r0)φ(e1)φ(r0)φ(r1) = E3EΣ4i=1αi

R3

(1.5.3)= E3RΣ4

i=1αi= φ(e1)φ(r0)φ(r1)φ(r0),

φ(e0)φ(r1)φ(e0) = E1E2(E4R3E4)E1E2

(1.5.3)= E1E2E4E2E1

(1.3.6)+(1.3.2)= δ2E1E2E4 = φ(δ

2e0).

Relations (1.6.19) and (1.6.23) under φ acting on generators hold by the above veri-fication for (1.6.18), (1.6.21) and the natural opposition involution (Remark 1.5.6) onBr(D4).The remaining relations can be proved similarly.

7.4 Normal forms of BrM(G2)

The following lemma can be shown to hold by an argument similar to the proof of 5.3.1.

Lemma 7.4.1. The submonoid of BrM(G2) generated by r0 and r1 is isomorphic to W (G2).

Lemma 7.4.2. The following equalities hold in Br(G2).

r0r1e0 = e1e0, (7.4.1)

e0r1r0 = e0e1, (7.4.2)

e0e1e0 = δ2e0, (7.4.3)

r1r0e1r0r1e0 = δe0. (7.4.4)

Proof. These equalities are derived as follows.

r0(r1e0)(1.6.18)= (r0r0)e1e0

(1.6.1)= e1e0,

e0r1r0(1.6.19)= e0e1r0r0

(1.6.1)= e0e1,

e0e1e0(7.4.1)= e0r0r1e0

(1.6.2)= e0r1e0

(1.6.22)= δ2e0,

r1r0e1(r0r1e0)(7.4.1)= r1r0e1e1e0

(1.6.3)= δr1r0e1e0

(1.6.18)= δr1r1e0

(1.6.1)= δe0.

108 TYPE G2

Remark 7.4.3. Consider (7.4.4). The image of the left side under φ is EX cl , and of theright side under φ is δEX , where

X = {α1,α2,α4}, X cl = {α1,α2,α4, 2α3 +α1 +α2 +α4}.

It is known from Section 1.3 that EX cl = δEX , this also serves for (7.4.4) holding underφ.

Recall (1.6.30).

Lemma 7.4.4. Let Di be a set of left coset representatives for Ni in W (G2) for i = 0, 1, andK0 = ⟨1⟩ ⊂ N0, K1 = ⟨r0r1r0r1r0⟩ ⊂ N1. Then for any r ∈ W (G2), there exist a ∈ Di andb ∈ Ki such that

rei = aei b.

Proof. From the proof of Lemma 1.6.6, we see that

r1r0r1r0r1e0 = e0,

r0r1r0r1r0e1 = e1r0r1r0r1r0.

Therefore our lemma holds by the analogous argument in Proposition 3.4.7.

Lemma 7.4.5. For any β ∈ Ψ+ and i ∈ {0,1}, there exist a1 a2 ∈ W (G2), j ∈ {0,1} andt ∈ Z such that

eβ ei = δt a1e ja2.

Proof. It is known that

Ψ+ = {α0, r1α0, r0r1α0,α1, r0α1, r1r0α1}.

We verify the lemma by distinguishing all possible cases.

e0e0 = δ3e0,

r1e0r1e0(1.6.22)= δr1e0,

r0r1e0r1r0e0(1.6.2)+(1.6.22)

= δr0r1e0,

e1e0(7.4.1)= r0r1e0,

r0e1r0e0(7.4.1)+(1.6.2)= r0r1e0,

r1r0(e1r0r1e0)(7.4.4)= δe0,

e0e1(7.4.2)= e0r1r0,

r1e0r1e1(7.4.2)+(1.6.2)= r1e0r1r0,

r0r1(e0r1r0e1)(7.4.2)+(1.6.3)= δr0r1e0r1r0,

e1e1 = δe1,

r0e1r0e1(1.6.23)= r1r0r1r0e1,

r1r0e1r0r1e1(1.6.23)+(1.6.2)

= r0r1r0e1.

7.5 THE ALGEBRA SBr(D4) 109

Lemma 7.4.6. Each element in BrM(G2) can be written in one of the following normal forms

(i) δkuei vw or

(ii) δka

where i = 0 or 1, u ∈ Di , v ∈ Ki , w ∈ Dopi , k ∈ Z, a ∈ W (G2). In particular, Br(G2) is

spanned by 39 elements over Z[δ±1].

Proof. It suffices to prove that the set of the normal forms is closed under multiplication.By Proposition 1.6.3 and Lemma 7.4.4, the product of two elements of the normal formsare still in the set of normal forms when one of the two elements belongs to the secondforms. If two elements δk1u1ei v1w1, δk2u2e j v2w2 are in the first forms, we have

δk1u1ei v1w1δk2u2e j v2w2

= δk1+k2u1v1w1u2((v1w1u2)−1ei v1w1u2)e j v2w2

∈ δkW (G2)etW (G2) Lemma 7.4.5.

Next we can apply Proposition 1.6.3 and Lemma 7.4.4 to conclude that the product is anormal form.The last claim follows from

#(W (G2)) +#(D0)2#(K0) +#(D1)

2#(K1) = 12+ 32 · 1+ 32 · 2= 39.

7.5 The algebra SBr(D4)

Now finish the proof of Theorem 7.1.1 by proving the surjectivity of φ.

Proof. First analogous to Section 6.5, it can be checked that the image φ(Br(G2)) is freeof rank 39 by the normal forms in Lemma 7.4.6 and by the normal forms of Br(D4) inTheorem 1.5.11 as follow.

(I) The restriction of φ to normal forms in (ii) of Lemma 7.4.6 is injective thanks tothe embedding of W (G2) to W (D4).

(II) Since there are three elements in the φ(W (G2))-orbit of {α1,α2,α4}cl; then the φon normal forms in (i) of Lemma 7.4.6 with i = 0 is an injective map to the normalforms in Theorem 1.5.11 associated to Y = {1,2, 4}.

(III) Since there are three elements in the φ(W (G2))-orbit of {α3} and we have 1 6=E3φ(r0r1r0r1r0)E3 ∈ W (M{3}); then the φ on normal forms in (i) of Lemma 7.4.6with i = 1 is an injective map to the normal forms in Theorem 1.5.11 associated toY = {3}.

We know the admissible root sets Y of type D4 involved for the normal forms in Theorem1.5.11 are

;, {3}, {1,2}, {1,4}, {1, 2,4}.

110 TYPE G2

If a ∈ SBrM(D4), we see that the monoid actions of a and aop on ; are σ-invariant.Because there is no σ-invariant element in the W (D4)-orbits of {α1,α2} and {α1,α4}, theonly possible Y for normal forms of elements in SBrM(D4) in Theorem 1.5.11 are

;, {3}, {1,2, 4}.

Therefore applying Theorem 1.5.11, we can verify that SBr(D4) is exactly equal toφ(Br(G2))described at the beginning of the proof and of rank 39 as in Section 4.5. This accom-plishes the proof of Theorem 7.1.1.

Now we have the decomposition for Br(G2) as Z[δ±1]-module as follows,

Br(G2) = Br(G2)/(r1r0e1r0r1)⊕ (r1r0e1r0r1)/(e0r1r0e1r0r1)⊕ (e0r1r0e1r0r1).

Similar to the arguments in [2] or Section 4.6, the theorem below about the cellularityfollows.

Theorem 7.5.1. If R is a field with characteristic not equal to 2 or 3, then Br(G2)⊗ R is acellularly stratified algebra.

Remark 7.5.2. From Chapter 5, we know that Br(I62) is of rank 2 · 6 + 3/2 · 62 = 66.

We see that {β0} is an admissible root set of type I62, which is not true for type G2.

This is the reason for the rank difference between Br(I62) and Br(G2). If δ = 1, there

exists a surjective homomorphism ϕ : Br(I62) → Br(G2) determined by ϕ(ri) = r1−i and

ϕ(ei) = e1−i , for i = 0, 1.

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INDEX

Br(M), 17Br(Q), 9Br(An), 9Br(Bn), 17Br(Cn), 17Br(Dn), 9Br(En), 9Br(F4), 17Br(G2), 17Br(Hk), 98Br(In

2), 87BrM(M), 17BrM(Q), 9BrM(Hk), 98SBr(A2n−1), 25SBr(D4), 105SBr(Dn+1), 49SBr(E6), 77an, 25f (n), 50BMW algebra, 23Brauer algebra of type M ∈ BCFG, 17Coxeter diagram, 5Coxeter group, 5Coxeter matrix, 5Coxeter system, 5Dynkin diagram, 6Weyl group, 6action of the Brauer monoid, 10admissible closure, 8, 20admissible orbit, 8admissible partition, 23admissible root set, 8admissible sets, 8admissible, 20cellularly stratified, 12cellular, 10

centralizer, 10connected component, 6connector, 54coroots, 6height, 6inflation, 11long root, 59negative roots, 7normalizer, 10orthogonal mate, 59root base, 6root datum, 6root system, 7roots, 6short root, 59simple root, 7symmetric submonoid, 50symmetric, 27

114

ACKNOWLEDGEMENTS

This thesis is my nice memory of four years study in the Discrete Algebra and Geometrygroup in Eindhoven University of Technology. It would not have come out without thehelp of many people.

First of all, I am most grateful to my supervisor, Arjeh Cohen for giving me the oppor-tunity to study in Eindhoven, letting me know many of his results and helping me finishmany papers and this thesis.

I want to express my gratitude to Andries Brouwer, Hans Cuypers, Eric Opdam, DavidWales for agreeing to serve on my Ph.D. committee and reading this manuscript andgiving many valuable comments, especially to David for his helpful suggestions for mypublished papers.

My special thanks go out not only to Shona Yu for being a co-author of some of mypapers and teaching me a lot about her algebras and categorification theory, but also toBart Frenk for his valuable knowledge in Brauer algebras and many interesting conver-sations. I also want to say thanks to Jan-Willem Knopper for helping me a lot in GAP andLatex.

I thank Max Horn for performing computations in GAP to explore the first few alge-bras and his enthusiastic support, and also thank Jan Draisma for his valuable suggestionsto solve some modeling problems.It is certain that our secretaries Anita Klooster and Rianne van Lieshout deserve a specialword of thanks for assisting me in many different ways.

I am very glad that other (graduated) Ph.D. students, Bart, Maxim, Rob, Yael, Cicek,Dan, Jos create a great working atmosphere in our group, and other Chinese students,Qingzhi, Xiulei, Yabin, Zhe, Lei, Wei, share many happiness in my daily life. In the Ph.D.life, I harvest my Ph.D. thesis with a lot of friendships and happy time.

Finally, I thank my family for their love, support and patience for me, especially formy wife, Di Li, making my last two years in Netherlands more enjoyable.

Shoumin Liu

Eindhoven, March 2012

116 ACKNOWLEDGEMENTS

CURRICULUM VITAE

Shoumin Liu wa born in Xintai, Shandong Province, China, on March 10, 1982. FromSeptember 2001, he started his bachelor degree at the School of Mathematical Scienceof Shandong University in Jinan. In July 2005, he obtained his bachelor degree with abachelor thesis in analytic number theory supervised by Prof Jianya Liu.

In September 2005, he started his master degree study at the Mathematics Instituteof Chinese Academy of Science in Beijing. During that academic year, he focused onnumber theory and algebraic geometry and was supervised by Prof Fei Xu. In 2006, hesuccessfully applied for the Algant Master Program of European Union. From September2006 to May 2007, he studied in the 1st university of Bordeaux in France with supervisorProf Qing Liu. From September 2007 to June 2008, he studied in Leiden University inNetherlands with supervisor Prof Hendrik Lenstra, where he finished his master thesisTrinomials and exponential Diophantine equations. In July 2008, he received three masterdegrees from the two European universities and the institute in China.

From 2008 until 2012 he was a Ph.D. student at Eindhoven University of Technology,under supervision of Prof. dr. A. M. Cohen. The present thesis is the result of his work inthis period.

118 CURRICULUM VITAE

SUMMARY:BRAUER ALGEBRAS OF

NON-SIMPLY LACED TYPE

Simple groups of Lie type and linear algebraic groups have a long history and play animportant role in mathematics. Mathematicians classify them by their Dynkin diagramsinto simply laced types and non-simply laced types. But these two kinds are not isolatedfrom each other. It is very classical knowledge that simple groups of Lie types of non-simply laced type can be obtained from the simply laced ones by considering nontrivialautomorphisms of their Dynkin diagrams. This idea was extended to a more generaltheory on admissible partitions by Mühlherr in the 1990s. In this thesis, we apply theseideas to the Brauer algebras of simply-laced type which are studied by Cohen, Frenk andWales and define Brauer algebras of non-simply laced type by generators and relations.We prove that they are free, compute their ranks and find bases by use of combinatorialdata on their root systems; we also prove their cellularity in the sense of Graham andLehrer. Our conclusions are summarized in the following table.

type rank BSO

Cn

∑ni=0

�

∑

p+2q=in!

p!q!(n−i)!

�22n−i (n− i)! A2n−1

Bn 2n+1 · n!!− 2n · n!+ (n+ 1)!!− (n+ 1)! Dn+1

F4 14985 E6

H3 1045 D6

H4 236025 E8

In2 (n≥ 5) 2n+ n2(n odd), 2n+ 3

2n2(n even) An−1

Here BSO abbreviates "being subalgebra of".In this thesis, we use many results of Brauer algebras of simply laced type, such as

their normal forms, their representations on collections of special mutually orthogonalroot sets, and their diagram representations. Also we use a lot of algebraic computationsto find normal forms for monomials in the algebras of non-simply laced type.

Documents

Brauer Algebras of Non-simply Laced TypeBrauer Algebras of Non-simply Laced Type PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag