Hecke Algebras and the Socle of the Projective Indecomposable Modules

192, 294]302 Ž1997. JA966967

JOURNAL OF ALGEBRA ARTICLE NO.

Hecke Algebras and the Socle of the Projective Indecomposable Modules Ahmed A. Khammash Department of Mathematics, Umm Algura Uni¨ ersity, P.O. Box 7296, Makkah, Saudi Arabia Communicated by Walter Feit Received April 22, 1996

0. INTRODUCTION This paper is an extension of wKx, in which we proved a criterion for determining the projective summands of permutation modules in terms of representations of their associated Hecke algebras. In this paper we extend that result to give the Žsimple. socle of such projective indecomposable summands. Let G be a finite group, let H F G, and let k be a field of characteristic r. Let Y s kGw H x, where w H x s Ý h g H h and write E s End k G Ž Y ., the endomorphism algebra of Y. E has a k-basis  a t ; t g T 4 indexed by a transversal set of  HgH; g g G4 , where a t g E is given by a t Žw H x. s Ý x g H t H x. Suppose that V is an indecomposable direct kG-summand of Y appears with multiplicity d Žnotation, w V < Y x s d ., and let S: E ª Md Ž k . be the corresponding irreducible matrix representation of E Žsee wG1, 3.4x.. For h g E, write SŽ h . s Ž s i j Ž h .. and write D Žs DSi, j . s Ý t g T si j Ž a t . ty1 g kG, for each pair Ž i, j . with 1 F i, j F d. Also write SocŽ V . for the socle of V which is, by definition, the maximal semisimple kG-submodule of V. It is well known that SocŽ V . is simple whenever V is projective. The main purpose of this paper is to prove the following theorem which links representations of the Hecke algebra EndŽ Y . with the kG-structure of Y. THEOREM 1. Let G, H, Y, D, and V be as abo¨ e. Then V is a projecti¨ e kG-module if and only if w H xD i, j w H x / 0 for some pair i, j, in which case SocŽ V . ( kGw H xD i, j w H x. 294 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

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The first statement of Theorem 1 was proved in wK, 1.4x. We shall give here another proof based on a functorial argument which is also vital to deduce the proof of the second statement. As applications of this theorem we are able to recover some Žpreviously known. results concerning the socle of some well-known modules. Namely, we determine the socle of the Steinberg module for finite groups of Lie type Ždetermined originally by N. Tinberg wTx.. A similar result is proved for the Specht modules for the symmetric group Žobtained originally by Farahat and Peel wFPx.. This work was inspired by the work of Green in wG1, G2, G3x. In fact the idea of this paper is an outcome of studying the functor q Y described in wG3x.

1. PRELIMINARIES In this section we give some functorial background which will be needed later. Let L be a finite dimensional k-algebra. We denote by mod L the category whose objects are all finite dimensional left L-modules. By Fun L we shall denote the category whose objects are all contravariant functors F: mod L ª mod k, where mod k is the category of vector spaces over k. For Y g mod L, write EŽ Y . for the endomorphism k-algebra End LŽ Y .. If F g Fun L and Y g mod L then F Ž Y . can be regarded as a right EŽ Y .module by the action ¨u [ F Ž u . Ž ¨ .

;u g E Ž Y . , ¨ g F Ž Y . .

This gives the evaluation functor e Y : Fun L ª mod EŽ Y . given by e Y Ž F . [ F Ž Y . for all F g Fun L. Write Žy, Y . Žg Fun L . for the hom functor: X ¬ Ž X, Y .L s Hom LŽ X, Y . for all X g mod L. For any right ideal R of EŽ Y ., let UY Ž R . be the subfunctor of Žy, Y . given by X ¬ UY Ž R .Ž X . [  f g Ž X, Y .L ; fg g R, ; g g Ž Y, X .4 . Adapting the argument of wG3, Sect. 3x, we find that R l UY Ž R . induces a bijection between the set of all maximal right ideals of EŽ Y . and the set of all maximal subfunctors Žy, Y . Žsee wG3, 3.8Žiii.x.. It follows that if Y is indecomposable, then JŽ EŽ Y .., the Jacobson radical of EŽ Y ., is the unique maximal ideal of EŽ Y . and so Žy, Y . has a unique maximal subfunctor r Žy, Y . [ UY ŽJŽ EŽ Y ... and so the quotient functor SY [ Žy, Y .rr Žy, Y . is simple. Write indec L for the subcategory of mod L whose objects are all indecomposable left L-modules. LEMMA 1.1 ŽAuslander wA, Sect. 2x.. Ž1. Any simple object in Fun L is isomorphic to SM for some M g indec L.

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Ž2. If X g mod L and M g indec L then SM Ž X . / 0 if and only if M is isomorphic to a direct L-summand of X, in which case SM Ž X . is a simple EŽ X .-module. Now suppose that n

Ys

Ý[ Yr

Ž 1.2.

rs1

is a direct sum decomposition of Y with Yr g indec L for all 1 F r F n. Let 1 EŽY . s Ýe r be the corresponding orthogonal primitive idempotent decomposition of 1 EŽY . in EŽ Y .. The following lemma relates the simple quotients of Žy, Y . to the simple functors SYr for at least one r, 1 F r F n. LEMMA 1.3. Let Y be as in Ž1.2.. If R is any maximal right ideal of EŽ Y ., there exists at least one 1 F r F n such that e r f R. For such an r, Žy, Y .rUY Ž R . is isomorphic to SYr . Proof. There must be some r with e r f R, or else we would have 1 EŽY . g R. Suppose that e r f R. Because R is a maximal right ideal of E, UY Ž R . is a maximal subfunctor of Žy, Y .. Therefore S s Žy, Y .rUY Ž R . is a simple functor. To prove that S ( SYr , it will be enough, by Lemma 1.1, to show that SŽ Yr . / 0. Let m r : Yr ª Y and pr : Y ª Yr be the inclusion and projection maps, so that m r pr s e r . From Lemma 1.1Ž1. we see that m r f UY Ž R .Ž Yr ., since m r pr f R. It follows that UY Ž R .Ž Yr . / Ž Yr , Y ., hence SŽ Yr . s Ž Yr , Y .rUY Ž R .Ž Yr . is not zero. Now if we take X sL L then S M Ž X . is nonzero if and only if M is projective indecomposable L-module. LEMMA 1.4. Suppose that M is a projecti¨ e indecomposable L-module. Then S M Ž L . ( Mrr M. In case L s kG Ž or if L is any symmetric algebra. Mrr M ( socŽ M ., so we ha¨ e S M Ž L . ( socŽ M .. Proof. Gabriel wGA, p. 4x shows that S M ( Žy, M .rŽy, r M ., hence S M Ž L . ( Ž L, M .rŽ L, r M .. If we apply the Žcovariant functor. Ž L, y. to the exact sequence 0 ª r M ª M ª Mrr M ª 0, we get a sequence 0 ª Ž L, r M . ª Ž L, M . ª Ž L, Mrr M . ª 0 which is exact because L is a projective L-module. Hence Ž L, M .rŽ L, r M . ( Ž L, Mrr M . ( Mrr M as required.

2. A USEFUL MORPHISM In this section we still take L to be any finite dimensional k-algebra and Y is any finite dimensional L-module. We define a morphism G: Žy, Y . ª DŽ Y, y. Žin the category Fun L ., where D is the dual functor, as

PROJECTIVE INDECOMPOSABLE MODULES

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follows: for each X g mod L, let G Ž X .: Ž X, Y . ª DŽ Y, X . be given by G Ž X . Ž f . s  g ¬ si j Ž fg . 4 gg Ž Y , X . ,

for all f g Ž X , Y . .

Here si j may be any coordinate function of the irreducible representation S of E s End LŽ Y . mentioned in the Introduction; later Žsee Lemma 2.4. we will choose i, j which are appropriate to a given summand Ž V s.Yr of Y. Now the kernel of G Ž X . is Ker G Ž X . s  f g Ž X , Y . ; si j Ž fg . s 0, ; g g Ž Y , X . 4 .

Ž 2.1.

This defines a subfunctor Ker G of Žy, Y .. We claim that Ker G s UY Ž R ., for some maximal right ideal R of E. Let U be an E-module affording S. That is, U is a k-space with basis u1 , u 2 , . . . , u d with E acting by d

ui h s

Ý si j Ž h . u j ,

for all i s 1, . . . , d and all h g E.

Ž 2.2.

js1

Take any i g  1, . . . , d4 . Define the right E-map w i : E ª U by setting w i Ž h. s u i h, h g E. w i is nonzero since si j is a coordinate function of an irreducible representation. Since U is simple, Im w i s U; hence R s ker w i is a maximal right ideal of E. The following lemma characterises Ker G Ž X . LEMMA 2.3. alent: Ža. Žb.

Let f g Ž X, Y .. Then the following two conditions are equi¨ -

si j Ž fg . s 0, ; g g Ž Y, X .. fg g R s Ker w i , ; g g Ž X, Y ..

Proof. Assume Ža. holds. Take any g g Ž Y, X .. Then gh g Ž Y, X . for all h g E, so by Ža. si j Ž fgh . s 0,

;h g E.

Ž ).

Let A s  p g E; si j Ž ph. s 0, ;h g E4 . Clearly A is a right ideal of E. Moreover, R : A, since if p g R, then ph g R, ;h g E, hence u i ph s 0, ;h g E, and so si j Ž ph. s 0, ;h g E, which implies that p g A. But R is a maximal right ideal, hence either A s R or A s E. We next show that A / E. If A s E, then 1 E g A, hence si j Ž h. s 0, ;h g E. But this implies, by Ž2.2., that u i h lies in the Ž d y 1.-dimensional k-space Ý[j/dis1 ku i , for all h g E. But this means that Im w i § U, which is a contradiction and so A s R. Hence, taking p s fg, clearly fg g A by Ž)., and so we have fg g R, ; g g Ž Y, X . and Žb. holds. Conversely if Žb. holds then, for any g g Ž Y, X ., w i Ž fg . s u i Ž fg . s 0 since fg g R. But then, by Ž2.2., u i Ž fg . s Ý j u i si j Ž fg . s 0, hence si j Ž fg . s 0 for all g g Ž Y, X ., which is Ža..

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Now Lemma 2.3 shows that Ker G Ž X . s UY Ž R .Ž X . Žsee Ž2.1.., and it follows that Ker G is equal to the maximal subfunctor UY Ž R . of Žy, Y .. Therefore Žy, Y .rKer G is a simple functor, and in fact we have PROPOSITION 2.4. Take any r g  1, 2, . . . , n4 such that Yr ( V Ž see the Introduction and Ž1.2... Choose i, j so that si j Ž e r . / 0. Then SV ( Žy, Y .rKer G ( Im G. Proof. It follows from si j Ž e r . / 0 that e r g R. By Lemma 1.3, Žy, Y .rUY Ž R . ( SYr . But Ker G s UY Ž R . and SYr ( SV, which proves that SV ( Žy, Y .rKer G. The isomorphism Žy, Y .rKer G ( Im G is standard.

3. PROOF OF THEOREM 1 In Sects. 1 and 2, L could be any finite dimensional k-algebra, and Y could be any finite dimensional left L-module. From now on, we assume that L s kG and Y s kGw H x as in the Introduction. In order to prove Theorem 1, it will be enough by Lemma 1.4 and Proposition 2.4 to show that Im G Ž kG . ( kG w H x D w H x .

Ž 3.1.

Let X g mod kG. There are standard natural isomorphisms

aX : Ž X , k . H ª Ž X , Y . G ,

bX : Ž k , X . H ª Ž Y , X . G

Ž 3.2.

Žsee wG2, p. 6x and wG3, p. 267x., where Žy, y.H s Hom k H Žy, y. and Žy, y. G s Hom k G Žy, y. and k is the trivial kG-module. So we have the following sequence: GŽ X .

DŽ Y , X . G

Db X

D Ž k, X . H ,

6

Ž X, Y .G

6

aX

6

Ž X, k. H

which is natural in X. This induces a natural transformation G1: Žy, k .H ª DŽ k, y.H defined by G1 Ž X . s Db X ( G Ž X . ( a X ,

for all X g mod kG.

Ž 3.3.

Since a X , b X are isomorphisms, Im G Ž X . ( Im G1Ž X .. So Theorem 1 will follow if we prove that Im G1 Ž kG . ( kG w H x D w H x .

Ž 3.4.

G1Ž X . takes j g Ž X, k .H to the element G1Ž X .Ž j . g DŽ k, X .H given by G1 Ž X . Ž j . Ž h . s si j Ž a X Ž j . b X Ž h . . ,

for all h g Ž k, X . H . Ž 3.5.

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PROJECTIVE INDECOMPOSABLE MODULES

To calculate a X Ž j . b X Žh ., which is an element of E s End k G Ž Y ., it is enough to see what it does to the element w H x of Y. This is done in wG2, p. 8x and in wG3, pp. 269, 270x. We find that

Ý j Ž ty1h Ž 1. . at

a X Ž j . bX Ž h . s

tgT

Žsee the Introduction for notation., hence si j Ž a X Ž j . b X Žh .. s j Ž Dh Ž1.., where D s Ý t g T si j Ž a t . ty1 . Thus by Ž3.5. we have G1 Ž X . Ž j . Ž h . s j Ž Dh Ž 1 . . ,

for all j g Ž X , k . H , h g Ž k, X . H .

Ž 3.6. Now we take X s kG Žsk G kG.. We have k-bases  j Z 4 , hZ 4 of Ž kG, k .H , Ž k, kG.H , respectively, each indexed by the set of H _ G of all right cosets Z s Hg in G. Namely, j Z : kG ª k takes g g G to 1 or zero, according as g g Z or not, and hZ : k ª kG is given by hZ Ž1. s w Z x Žcf. wK, p. 731x.. The space Ž kG, k .H and Ž k, kG.H are, respectively, left and right kG-modules, with G-actions as follows: g j Z s z Z g y 1 and

hZ g s hZ g ,

for all g g G, Z g H _ G. Ž 3.7.

It is clear that Ž kG, k .H s kGj H , hence Im G1Ž kG. s kG. P, where P s G1Ž kG.Ž j H .. By Ž3.6. and using wK, Lemma 1.3x, we have for all Z s Hg g H_G P Ž hH g . s j H Ž D w Hg x . s the coefficient of w H x in w H x D w H x g , where D s DŽ i , j. for a suitable Ž i , j . .

Ž 3.8.

We can write w H xDw H x s Ý t g T l t w HtH x, for some l t g k. The coefficient of w H x in w H xDw H x g s Ý t g T l t w HtH x is l u , where u is the unique element of T such that gy1 g HuH. But this is the same as the coefficient of w H x in g Ý t g T l t w HtH x, and this is the coefficient of gy1 w H x in w H xDw H x. So by Ž3.8., we have, for all g g G, P Ž hH g . s the coefficient of gy1 w H x in w H x D w H x .

Ž 3.9.

The k-space Y s kGw H x has basis w Zy1 x; Z g H _ G4 . There is a left kG-isomorphism c : kGw H x ª DŽ k, kG.H which can be defined as follows:

c Ž w Zy1 x . Ž hH g . s

½

1 0

if Z s Hg , if Z / Hg .

Ž 3.10.

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AHMED A. KHAMMASH

It follows from Ž3.10. that, for all g g G, c Žw H xDw H x.ŽhH g . s the coefficient of gy1 w H x in w H xDw H x; then by Ž3.9. it follows that c Žw H xDw H x. s P. Therefore Im G1Ž kG. ( kG. P ( kGw H xDw H x, i.e., Ž3.4. holds, and so Theorem 1 is proved. Remark. Since w H xDw H x s Ý t g T si j Ž a t .< H lt H
4. APPLICATIONS In this section we apply Theorem 1 to determine the socle of the Steinberg module for a finite group of Lie type and the socle of the Specht module for the symmetric groups. This will be done by embedding those two modules in certain projective indecomposable modules. But before doing that let us consider the following motivation example. A MOTIVATION EXAMPLE. The Hecke algebra E s End k G Ž Y . has a one-dimensional multiplicative character IND given by INDŽ a t . s < H lt H < for all t g T. The indecomposable component of Y which corresponds to IND is the Alperin]Scott module SŽ G, H . which is, by definition, the unique component of Y having the trivial kG-module k G in its head and its socle Žsee, for example, wG2, Sect. 4x.. Now

w H x D w H x s Ý IND Ž at . < H lt H < w HtH x tgT

s

Ý < H < w HtH x s < H < Ý t

ž

ggG

g

/

This implies, by Theorem 1, that SŽ G, H . is projective if and only if < H < / 0, in which case SocŽ SŽ G, H .. s kGw H xDw H x s kGŽÝ g g G g . ( k G as one should expect. The Steinberg Module Now let G s Ž G, B, N, R, U . be a finite group with split BN-pair in the sense of wCR, Sect. 69x with a Coxeter system ŽW, R ., where W s NrB l N. Take Y s kGwU x. Since G s D ng N UnU, we may take T s U and so E has a k-basis  a n : n g N 4 . E has a multiplicative character C given by C Ž a n . s Žy1. l Žp Ž n.. , where p : N ª W is the natural map and l is the length function of W wTx. The indecomposable component of Y which corresponds to C is denoted by YC and is called the Steinberg component

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301

of G. The Steinberg module of G, as defined originally by R. Steinberg wSx, is the ideal StG s kG N 4wU x, where  N 4 s Ý ng N Žy1. l Žp Ž n.. n. It is known Žsee wTx. that StG is a kG-submodule of YC . This implies that SocŽ StG . s SocŽ YC .. PROPOSITION 4.1 ŽTinberg wTx.. SocŽ StG . s kGwU x N 4wU x. This holds in any characteristic of k. Proof. The Weyl group W of G has a unique element w 0 of maximal length. Let n 0 g N be such that p Ž n 0 . s w 0 . Then < U ln 0 U < s 1 and C Ž a n 0 . s Žy1. l Ž w 0 . / 0, hence C Ž a n 0 .< U ln 0 U < / 0. This implies that wU xDwU x / 0, whatever the characteristic of k. Hence, by Theorem 1, YC is always projective with Soc Ž YC . Ž s Soc Ž StG . . s kG w U x  N 4 w U x , since D s  N 4 . The Specht Module Now take G s S n , the symmetric group on n letters. Let l be a partition of n and take H s R lŽ K s Cl . to be the row Žcolumn. stabilizer of l. Let Y s kGw H x and T be a transversal for  HnH; n g G4 . Green wG2x proved that the Hecke algebra E s End k G Ž Y . has a multiplicative character Cl: E ª k given by ClŽ a t . s Ý x g H t H l K « Ž x . for all t g T, where « is the sign character of G. The indecomposable kG-summand of Y which corresponds to Cl is the Young module of type l and is denoted by Yl. The Specht module of G of type l is the ideal S l s kG K 4w H x, whee  K 4 s Ý x g K « Ž x . x. It is known Žsee wJ, Sect. 8x. that S l is a kG-submodule of Yl. The following determines the socle of S l for r-regular partitions l of n. PROPOSITION 4.2 Žsee wFP, Theorem 2.3x.. partition of n. Then SocŽ S l . ( kGw H x K 4w H x.

Suppose l is an r-regular

Proof. Since S l F Yl, SocŽ S l .. On the other hand, Y l is projective whenever l is r-regular ŽwK, 3.2x.. So, by Theorem 1, SocŽ Yl . s kGw H xDw H x, where D s Ý x g T ClŽ a x . x. But since C Ž a x . s 0 for any double coset HxH which does not meet K, we can arrange that x g K whenever HxH l K / B. We then have

w H xDw H x s Ý C Ž ax . w H x x w H x s

Ý « Ž x. w H x xw H x s w H x K 4 w H x.

Hence SocŽ S l . ( kGw H x K 4w H x.

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ACKNOWLEDGMENTS This paper was written while the author was holding a visiting position at the University of Oregon. Thanks to Professor Charles Curtis for many useful discussions during that visit. Many thanks also to Professor J. A. Green for his comments on the subject of this paper.

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M. Auslander, Representation theory of artin algebras II, Comm. Algebra I Ž1974., 269]310. C. Curtis and I. Reiner, ‘‘Methods of Representation Theory II,’’ Wiley-Interscience, New York, 1985. K. Farahat and M. Peel, On representation theory of the symmetric groups, J. Algebra 67 Ž1980., 280]304. P. Gabriel, ‘‘Auslander]Reiten Sequence and Representation Theory I,’’ Lecture Notes in Mathematics, vol. 831, Springer-Verlag, Berlin, 1980. J. A. Green, Multiplicities, Scott modules and lower defect groups, J. London Math. Soc. Ž 2 . 28 Ž1983., 282]292. J. A. Green, Functor categories and group representations, Portugal. Math. 43 Ž1986., 9]22. J. A. Green, On three functors of M. Auslander’s, Comm. Algebra 15Ž1 & 2. Ž1987., 241]277. G. James, The representation theory of the symmetric groups, Proc. Sympos. Pure Math. 47 Ž1987.. A. Khammash, Functors and projective summands of permutation modules, J. Algebra 163Ž3. Ž1994., 729]738. R. Steinberg, Prime power representation of finite linear groups 2, Canad. J. Math. 9 Ž1957., 347]351. N. Tinberg, The Steinberg component of finite group with split BN-pair, J. Algebra 61 Ž1979., 508]526.