Ž .JOURNAL OF ALGEBRA 202, 192]201 1998ARTICLE NO. JA977299

The Loewy Structure and Basic Algebra Structure forSome Three-dimensional Projective Special Unitary

Groups in Characteristic 3

Katrina Hicks

The Uni ersity of Western Australia, Nedlands, Western Australia 6907

Communicated by Walter Feit

Received February 24, 1997

1. INTRODUCTION

The purpose of this paper is to determine the basic algebra structure forblocks of maximal defect for some of the projective special unitary groupsover an algebraically closed field of characteristic three. We concentrateon the projective special unitary groups in three dimensions over a field of

2 Ž .q elements q ) 2 where q is a prime power with q ' 2 mod 3 andq k 8 mod 9. These are all simple groups in nondefining characteristic withcommon Sylow 3-subgroup C = C , the elementary abelian group of3 3order 9.

Žw x. Ž 2 .Geck 4 studied the Brauer characters for SU q , the special unitary3group over a field of q2 elements, for all odd prime characteristics notdividing q. In particular in the case 3rq q 1, the decomposition numbersfor the principal block were determined up to the presence of a scalar

Žwhich is dependent on the power of 3 dividing q q 1 all other blocks have.defect zero or one . We consider the case in which 3rq q 1 but 9 does not,

i.e., q ' 2 mod 3 and q k 8 mod 9, and we use the information determinedby Geck to investigate the representation theory of the principal block ofthe corresponding projective special unitary groups, this being the onlyblock of maximal defect for these groups. In this paper we determine theLoewy and socle structure of the principal indecomposable modules of theprincipal block. This information can then be used to determine a quiverand relations defining a 2-parameter family of algebras to which the basic

Ž w x.algebras must belong see 5 . The main results are:

192

0021-8693r98 $25.00Copyright Q 1998 by Academic PressAll rights of reproduction in any form reserved.

LOEWY AND BASIC ALGEBRA STRUCTURES 193

THEOREM 1.1. Suppose q ) 2 is a prime power with q ' 2 mod 3 andq k 8 mod 9. The Loewy structures and socle structures of the PIMs in the

Ž .principal block of U q o¨er an algebraically closed field K of characteristic 33are as follows,

1 A BD D D

A B C B 1 C C 1 AD D D1 A B

C DD 1 A B C

A 1 B D D DD 1 A B CC D

Ž . Ž .where 1 is the tri ial KU q -module, D is a simple KU q -module of3 32 Ž .dimension q y q o¨er K, and A, B, and C are simple KU q -modules of3

Ž .Ž 2 .dimension q y 1 q y q q 1 r3 o¨er K.

THEOREM 1.2. Suppose q ) 2 is a prime power with q ' 2 mod 3 andŽ .q k 8 mod 9. The basic algebra of the principal block of U q o¨er an3

algebraically closed field K of characteristic 3 is determined by the qui er inFigure 1 and the relations

a Xa s 0, g Xg s 0,

b Xb s 0, d Xd s 0,

l y 1 aa X q yl bb X q gg X q l 1 y l dd X s kYdd Xaa X ,Ž . Ž . Ž .bb Xd s aa Xd , d Xbb X s d Xaa X ,

gg Xd s aa Xd , d Xgg X s d Xaa X ,

bb Xa s dd Xa , a Xbb X s a Xdd X ,

gg Xa s l2dd Xa , a Xgg X s l2a Xd X ,

aa Xb s dd Xb , b Xaa X s b Xdd X ,2 2X X X X X Xgg b s l y 1 dd b , b gg s l y 1 b dd ,Ž . Ž .

aa Xg s l2dd Xg , g Xaa X s l2g Xdd X ,2 2X X X X X Xbb g s g y 1 dd g , g bb s l y 1 g dd ,Ž . Ž .

where l and kY are unknown scalars from the field K. That is, the qui er andrelations determine a 2-parameter family of algebras to which the basic algebraof the principal block must belong.

KATRINA HICKS194

FIGURE 1

Ž .Throughout this paper we let U q denote the three-dimensional pro-32 Ž .jective special unitary group over a field of q elements q ) 2 where

q ' 2 mod 3 and q k 8 mod 9, we let K denote an algebraically closedŽ .field of characteristic three and we let KU q denote the corresponding3

group algebra. All modules are acted upon on the left and are finitedimensional over K.

2. NOTATION AND PRELIMINARY RESULTS

For any finite group G, we denote the radical of the left KG-module MŽ .by Rad M and we let

Radiy1 MŽ .L M sŽ .i iRad MŽ .

Ž .denote the ith Loewy layer of M. The Loewy length, ll M of M is theiŽ .smallest value of i for which Rad M s 0. The Loewy structure of M is a

Ž .diagram whose ith layer downward gives the simple summands of L Miwith multiplicities. We denote the largest semisimple submodule of M, the

Ž . Ž .socle, by S M and we let S M be defined recursively for i ) 1 by1 iŽ .choosing S M to be the module satisfyingi

S M MŽ .i ( S ,1 ž /S M S MŽ . Ž .iy1 iy1

Ž . Ž .so that we have a series 0 : S M : S M : ??? : M called the socle1 2Ž . Ž .series. Then S M rS M is called the ith socle layer of M. Thei iy1

Ž .smallest value of i for which S M s M is called the socle length of MiiŽ . Ž .and is equal to the Loewy length. Note that Rad M : S M for allll ŽM .y i

i G 0. The socle structure of M is a digram whose ith layer upward givesthe simple summands of the ith socle layer with multiplicities. The first

LOEWY AND BASIC ALGEBRA STRUCTURES 195

Loewy layer will often be referred to as the head of M, while we will referŽ . Ž . Ž . Ž . Uto Rad M rRad M l S M as the heart of M, denoted by Ht M . M1

will denote the contragradient dual of M, and P will denote the projec-Mtive cover of M.

The following tools for dealing with Loewy structures and socle struc-tures will be of fundamental importance:

Žw x .LEMMA 2.1 7 Lemma 8.4 . The socle structure of M is the dual of theLoewy structure of MU.

Žw x . Ž .LEMMA 2.2 7 Lemma 8.6 . M contains a simple module T in L M ifiand only if M has a factor module of Loewy length i with simple socle T.

Žw x .LEMMA 2.3 7 Lemma 9.10 . Let M and N be simple KG-modules. ThenŽ . Ž .the multiplicity of M in L P is the same as the multiplicity of N in L P .i N i M

We have the following theorem for group algebras.

Žw x . Ž .THEOREM 2.4 8 Theorem E . Let Ht P denote the heart of the1projecti e co¨er of the tri ial KG-module and let P be a Sylow p-subgroup of

Ž .G. If P is not a dihedral 2-group then Ht P is indecomposable.1

Ž .Let V M denote the kernel of an epimorphism of the projective covery1Ž .of M to M, and let V M denote the cokernel of an embedding of M

into the injective hull of M. We call a KG-module M V-periodic if there isan exact sequence,

0 ª M ª P ª ??? ª P ª P ª M ª 0,ny1 1 0

with the P projective. This is equivalent to saying there exists n ) 0 and ainŽ .projective module P with M ( V M [ P. We have the following useful

theorem for determining when a module over a group algebra is notperiodic:

Žw x .THEOREM 2.5 1 Theorem 3 . If G has p-rank greater than 1, then e¨eryV-periodic KG-module has dimension di isible by p.

We will also make use of some Auslander]Reiten theory. For a generalw xcoverage of Auslander]Reiten theory see, for example, 2 . For any non-

projective indecomposable module M, there is an almost split sequenceterminating at M and for a symmetric algebra this sequence has the form,

0 ª V2 M ª E ª M ª 0.Ž .

The nonprojective direct summands of E are exactly the modules X in the2Ž .stable Auslander]Reiten quiver for which there are arrows V M ª X

and X ª M. For a simple nonprojective module S we have an Auslan-

KATRINA HICKS196

der]Reiten sequence,

0 ª V S ª P [ Ht P ª Vy1 S ª 0,Ž . Ž . Ž .S S

Ž . Ž . y1Ž .so that Ht P determines the modules incident with V S and V S inSthe stable Auslander]Reiten quiver.

For a group algebra KG, the structure of a connected component of thestable Auslander]Reiten quiver takes on a limited number of forms. Forour purposes we make use of the following:

w x w xTHEOREM 2.6 3 , 6 . Let K be an algebraically closed field and let P be aSylow p-subgroup of G which is not cyclic, dihedral, semidihedral, or general-ized quaternion. Suppose Q is a connected component of the stable Auslan-der]Reiten qui er and suppose Q contains an indecomposable module whoseK-dimension is not di isible by p. Then Q is isomorphic to Z A where Z A is` `

Ž .the graph see Fig. 2 .

3. THE LOEWY AND SOCLE STRUCTURES

The decomposition matrix and Cartan matrix for the principal block ofŽ .U q , which were determined from the information presented by Geck3

Žw x.4 may be found in the appendix. The principal block has five simplemodules, the trivial module, which we denote by 1, a module of dimension

2 Ž .Ž 2q y q, which we denote by D, and three modules of dimension q y 1 q.y q q 1 r3, which we denote by A, B, and C. All five simple modules are

self-dual and A, B, and C are conjugate under a diagonal automorphismŽ .of U q while the involutory field automorphism fixes A and conjugates3

B and C.P has composition factors 2.1 q 2.D q 1. A q 1.B q 1.C. P is self-dual1 1

and invariant under the diagonal automorphism so that A, B, and C mustall lie in the same Loewy layer. We also know from Theorem 2.4 that the

FIGURE 2

LOEWY AND BASIC ALGEBRA STRUCTURES 197

heart of P is indecomposable and therefore the Loewy structure of P is1 1

1D

.A B CD1

P has composition factors 1.1 q 2.D q 2. A q 1.B q 1.C and is invari-Aant under the field automorphism, so that B and C must lie in the sameLoewy layer. Furthermore

dim Ext1 A , 1 s dim Ext1 1U , AU s dim Ext1 1, A s 0.Ž . Ž . Ž .K K K

P is also self-dual, as are all the simple composition factors, and so theAonly possibilities for the Loewy structures of P areA

A AD B D C

and .B 1 C 1D DA A

In the latter case the heart is decomposable and we have an almost splitsequence,

Dy10 ª V A ª P [ [ B [ C ª V A ª 0,Ž . Ž .1A ž /D

so that the component of the stable Auslander]Reiten quiver containingŽ .V A has arrows as indicated in Figure 3. The dimension of B is

Ž .Ž 2 . Ž 2 .q y 1 q y q q 1 r3 and q y q q 1 is not divisible by 9, thereforethe dimension of B is not divisible by 3. By Theorem 2.6 the component is

FIGURE 3

KATRINA HICKS198

Ž .isomorphic to Z A and so there are at most two arrows starting at V A ,`

which contradicts the existence of three indecomposable summands in theheart of P . Hence the Loewy structure of P isA A

AD

.B 1 CDA

Given the action of the diagonal automorphism the Loewy structures of PBand P areC

B CD D

and .C 1 A A 1 BD DB C

P has composition factors 2.1 q 5.D q 2. A q 2.B q 2.C and is self-Ddual, as are all the composition factors, and applying Lemma 2.3 the onlypossible Loewy structures are

D1 A B C D

D1 A B C

DD

iŽ .

D D1 A B C D D 1 A B C D

D D1 A B C D 1 A B C

D D

ii iiiŽ . Ž .

D D1 A B C D 1 A B C

D D D D D1 A B C 1 A B C .

D D

iv vŽ . Ž .

LOEWY AND BASIC ALGEBRA STRUCTURES 199

Ž .Suppose P has the Loewy structure given in i and consider theD2Ž .module M s Rad P . Then M is a quotient of P of Loewy length 4D D

Ž . Ž .and as M : P , Soc M : Soc P s D. Therefore, by Lemma 2.2, PD D Dwould have a composition factor isomorphic to D in the fourth Loewylayer, which is not the case.

Ž .For case ii we have that the heart of P has Loewy and socle structure,D

1 A B C D,D

1 A B C D

and so we have a submodule with Loewy and socle structure,

1,D

1 A B C D

and the only quotient of P with two composition factors isomorphic to 11Ž .is P itself. Hence case ii is not possible.1

Ž . Ž .For cases iii and iv the heart of P is decomposable with anDindeomposable summand isomorphic to D. Therefore the component of

Ž .the stable Auslander]Reiten quiver containing V D has an arrow D ªŽ . Ž .V D , so that D and V D are in the same component of the stable

Auslander]Reiten quiver. Hence the Heller operator V induces a graphautomorphism of the component containing D. Because D is an indecom-posable module of dimension q2 y q, which is not divisible by 3, thiscomponent is isomorphic to Z A by Theorem 2.6. Let U be a module`

corresponding to a vertex in the first row of Z A . Then our component has`

Ž 2the form in Figure 4, because V acts as a shift one to the left for any.component of the stable Auslander]Reiten quiver. Because V induces a

graph automorphism, it must fix the modules in the first row, i.e., thoseŽ .vertices with only one arrow starting and ending at them, and so V U (

2 kŽ .V U for some integer k ) 0. Hence U is V-periodic and in fact every

FIGURE 4

KATRINA HICKS200

module in the first row is V-periodic. Now let W be a module in thesecond row. Then there is some module U in the first row such that thiscomponent of the stable Auslander]Reiten quiver has arrows

2 ky1Ž .V W66

2 ky1 2 ky3Ž . Ž .V U V U ,

2 ky1Ž .so that in fact W ( V W . Hence every module in the second row isV-periodic. By applying an inductive argument, we find that every modulein this component is V-periodic, in particular, D is V-periodic. However,by Alperin’s result on V-periodicity, Theorem 2.5, this means that thedimension of D must be divisible by 3, which is not the case. Hence Dcannot be a direct summand of the heart of P as claimed. Therefore, theDonly possibility left for the Loewy structure of P isD

D1 A B C

.D D D1 A B C

D

We have therefore determined the Loewy structures of all the PIMs, andas they are all self-dual, applying Lemma 2.1 completes the proof ofTheorem 1.1.

APPENDIX

Ž .The decomposition matrix for U q is3

x 1 0 0 0 01

2x 0 1 0 0 0q yq

Ž0.2x 0 0 1 0 0Žqy1.Ž q yqq1.r3

.Ž1.2x 0 0 0 1 0Žqy1.Ž q yqq1.r3

Ž2.2x 0 0 0 0 1Žqy1.Ž q yqq1.r3

3x 1 2 1 1 1q

LOEWY AND BASIC ALGEBRA STRUCTURES 201

Ž .The Cartan matrix for U q is3

1 D A B C1 2 2 1 1 1D 2 5 2 2 2 .A 1 2 2 1 1B 1 2 1 2 1C 1 2 1 1 2

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