JOURNAL OF ALGEBRA 194, 496᎐520Ž. 1997 ARTICLE NO. JA966929
Semisimplicity and Tensor Products of Group Representations: Converse Theorems
Jean-Pierre Serre
College` de France, 3 rue d’Ulm, F-75005 Paris
With an appendix by Walter Feit
Department of Mathematics, Yale Uni¨ersity, New Ha¨en, Connecticut 06520-8283
Communicated by Efin Zelmano¨
Received October 8, 1996
INTRODUCTION
Let k be a field of characteristic p G 0, and let G be a group. If V and W are finite-dimensional G-modules, it is known that:
Ž.1 V and W semisimple « V m W semisimple if p s 02,p.88,Žwx .or if p)0and dim V q dim W - p q 2Žwx 6 , Corollary 1 to Theorem 1..
2 Ž.2 V semisimple « H V semisimple if p s 0 or if p ) 0 and dim V F Ž.Žpq3r2 cf.wx 6 , Theorem 2. .
We are interested here in ‘‘converse theorems’’: proving the semisimplic- 2 ity of V from that of V m W or of H V. The results are the following Ž.cf. Sects. 2, 3, 4, 5 :
Ž.3 VmW semisimple « V semisimple if dim W k 0 Ž mod p .. m Ž.4 mV semisimple « V semisimple if m G 1. 2 Ž.5HV semisimple « V semisimple if dim V k 2Ž. mod p . 2 Ž.6 Sym V semisimple « V semisimple if dim V k y2Ž. mod p . m Ž.7HV semisimple « V semisimple if dim V k 2, 3, . . . , m Ž.mod p . 496
0021-8693r97 $25.00 Copyright ᮊ 1997 by Academic Press All rights of reproduction in any form reserved. GROUP REPRESENTATIONS 497
Examples show that the congruence conditions occurring inŽ.Ž.Ž. 3 , 5 , 6 , andŽ. 7 cannot be suppressed: see Sect. 7 for Ž. 3 , Ž. 5 , and Ž. 7 and the Appendix forŽ. 5 and Ž. 6 . These examples are due to Ž or inspired by . W. Feit.
1. NOTATION
1.1. The Category CG As in the Introduction, G is a group and k is a field; we put charŽ.k s p. The category of kGwx-modules of finite dimension over k is denoted by CGGG.If Vand W are objects of C , the k-vector space of C -morphisms of Vinto W is denoted by HomGŽ.V, W .
1.2. Split Injections
A CG-morphism f: V ª W is called a split injection if there exists a left inverse r: W ª V which is a CG-morphism. This means that f is injective, and that its image is a direct factor of W, viewed as a kGwx-module. We also say that f is split. If f: V12ª V and g: V 23ª V are split injections, so is g ( f. Con- versely, if g ( f is a split injection, so is f. An object W of CG is semisimple if and only if every injection V ª W is split.
1.3. Tensor Products
The tensor productŽ. over k of two objects V and VЈ of CG is denoted by V m VЈ. If V ª W and VЈ ª WЈ are split injections, so is V m VЈ ª W m WЈ. The vector space k, with trivial action of G, is denoted by 1. We have 1 m V s V for every V.
1.4. Duality
The dual of an object V of CG is denoted by V *. If W is an object of Ž. Ž. CGk, one has W m V * s Hom V, W , the action of G on Hom kV, W y1 Ž. Ž being f ¬ sfs for s g G. An element f of Hom k V, W is G-linear i.e., belongs to HomGŽ..V, W if and only if it is fixed under the action of G. Ž. In particular, one has V m V * s End kVV . The unit element 1 of Ž. End kVV defines a G-linear map i :1ªVmV*, which is injective if V/0. 498 SERRE AND FEIT
1.5. Trace
The trace tV : V m V * ª 1isaG-linear map. The composite map
tVV(i :1ªVmV*ª1 G is equal to dim V, viewed as an element of k s EndŽ. 1 ; it is 0 if and only if dim V ' 0Ž.Ž mod p . When p s 0 this just means dim V s 0..
2. FROM V m W TO V
Let V and W be two objects of CG.
PROPOSITION 2.1. Let VЈ be a subobject of V. Assume:
iW:1ªWmW* is a split injection,Ž. 2.1.1 and VЈ m W ª V m W is a split injection.Ž. 2.1.2 Then VЈ ª V is a split injection. Proof. Consider the commutative diagram:
Ј 6 VЈ VЈ m W m W *
␣ ␥ 6 6
 6 V V m W m W *, where the vertical maps come from the injection VЈ ª V and the horizon- Ž.Ž. tal maps are  s 1VWm i and Ј s 1VЈm iWcf. Sect. 1.5 . By 2.1.1 , Ј is split; byŽ. 2.1.2 , VЈ m W ª V m W is split, and the same is true for ␥. Hence  ( ␣ s ␥ ( Ј is split, and this implies that ␣ is split. Remark 2.2. AssumptionŽ. 2.1.1 is true in each of the following two cases:
Ž.2.2.1 When dim W k 0 Ž mod p ., i.e., when dim W is invertible in k. Indeed, if c denotes the inverse of dim W in k, the map
c и tW : W m W * ª 1 is a left inverse of iW Ž.cf. Sect. 1.5 . Ž.2.2.2 When W / 0 and W m W * is semisimple, since in that case every injection in W m W * is split. PROPOSITION 2.3. Assume Ž.2.1.1 and that V m W is semisimple. Then V is semisimple. GROUP REPRESENTATIONS 499
Proof. Let VЈ be a subobject of V. Since V m W is semisimple, the injection VЈ m W ª V m W is split, henceŽ. 2.1.2 is true, and Proposition 2.1 shows that VЈ ª V splits. Since this is true for every VЈ, it follows that V is semisimple. Alternate proofŽ. sketch . One uses Ž. 2.1.1 to show that the natural map
n n H Ž.ŽG, Hom kŽ.V12, V ª H G, Hom kŽ.V12m W, V m W .
is injective for every n, V12, V .If V is an extension of V1by V 2and Ž.V Ž. denotes the corresponding element of the group Ext V12, V s 1ŽŽ.. HG, Hom k V12, V , the assumption that V m W is semisimple implies Ž. Ž . Ž. that V gives 0 in Ext V12m W, V m W and hence V s 0. This shows that V is semisimple.
THEOREM 2.4. If V m W is semisimple and dim W k 0Ž. mod p , then V is semisimple. Proof. This follows from Proposition 2.3 and RemarkŽ. 2.2.1 .
Remark. The condition dim W k 0Ž. mod p of Theorem 2.4 cannot be suppressed. This is clear for p s 0, since it just means W / 0; for p ) 0, see Feit’s examples in Sect. 7.2.
n m 3. FROM T V m T V *TOV
Let V be an object of CG.
LEMMA 3.1. The injection jVVVs 1 m i : V ª V m V m V * is split. Ž. Proof. If we identify V m V * with End k V , the map
jVk: V ª V m End Ž.V
Ž. is the map x ¬ x m 1VV . Let f : V m End kV ª V be the ‘‘evaluation Ž.Ž Ž .. map’’ x ¬ xxgV,gEnd kVV . It is clear that f ( jVs 1V . Hence jV is a split injection. n If n G 0, let us write T V for the tensor product V m V m иии m V of n 0 copies of V, with the convention that T V s 1. PROPOSITION 3.2. Let VЈ be a subobject of V. Assume that the natural n n 1 n 1 injection of T VЈ s VЈ m T y VЈ in V m T y VЈ splits for some n G 1. Then VЈ ª V splits. 500 SERRE AND FEIT
Proof. This is clear if n s 1. Assume n G 2, and use induction on n. We have a commutative diagram:
n 1 6 n 2 T y VЈ V m T y VЈ
␥  6 6
6 n 1 n 2 T y VЈ m VЈ m VЈ* V m T y VЈ m VЈ m VЈ*, where the horizontal maps are the obvious injections, and the vertical ones Ž. are of the form x ¬ x m 1V Ј , with 1V Ј g VЈ m VЈ* cf. Sect 1.4 . ny2 If we put W s T VЈ, we may write ␥ as 1WVm j Ј, where jVЈis the map of VЈ into VЈ m VЈ m VЈ* defined in Lemma 3.1Ž with V replaced by VЈ.. From this lemma, and from Sect. 1.3, it follows that ␥ is a split injection. On the other hand, is the tensor product of the natural n n 1 injection T VЈ ª V m T y VЈ, which is split by assumption, with the identity map of VЈ*; hence is split and the same is true for  ( s (␥, hence also for . By the induction assumption this shows that VЈ ª V is a split injection.
n m THEOREM 3.3. Assume that T V m T V * is semisimple for some inte- gers n, m G 0, not both 0. Then V is semisimple.
n COROLLARY 3.4. If T V is semisimple for some n G 1, then V is semisimple.
Proof of Theorem 3.3. Consider first the case of Corollary 3.4, i.e., n 1 m s 0, n G 1. Let VЈ be a subobject of V. Then V m T y VЈ is a n n subobject of T V. Since T V is assumed to be semisimple, so is V m n 1 n n 1 T y VЈ. Hence the injection T VЈ ª V m T y VЈ splits. By Proposition 3.2, this implies that VЈ ª V splits. Since this is true for every VЈ,it follows that V is semisimple. Since duality preserves semisimplicity, the same result holds when n s 0 and m G 1. Hence, we may assume that n G 1 and m G 1, and also that V / 0. If n and m are both equal to 1, then V m V * is semisimple by assumption. Put W s V *; usingŽ. 2.2.2 we see that W has property Ž. 2.1.1 and by Proposition 2.3 this implies that V is semisimpleŽ I owe this argument to W. Feit. . The remaining case n q m G 3 is handled by n 1 m 1 induction on n q m, using the fact that T y V m T y V * embeds into n m T V m T V *, hence is semisimple. GROUP REPRESENTATIONS 501
2 2 4. FROM H V AND Sym V TO V
4.1. Notation
Let V be an object of CGV, and let be the canonical map
2 VmVªHV.
2 Define V : V ª H V m V * as the composite of the maps jV : V ª V m V Ž. 2 mV* cf. Lemma 3.1 and VVm 1:*VmVmV*ªHVmV*. Define 2 V :HVmV*ªVas the composite
2 H V m V * ª V m V m V * ª V, where the map on the left is Ž.x n y m z ¬ x m y m z y y m x m z Žfor . x, y g V, z g V * and the map on the right is the map fV defined in the proof of Lemma 3.1, i.e., x m y m z ¬ ²:x, zy. We have
²: ²: VŽ.Ž.xnymzsx,zyy y,zxŽ x,ygV,zgV*..
Both VVGand are C -morphisms.
PROPOSITION 4.2. The composite map
6 6 VV2 VHVmV*V
Ž. is equal to 1 y n 1,V where n s dim V. U Proof. Choose a k-basis Ž.e␣ of V, and let Že␣ .be the dual basis of V *. U We have 1V s Ýe␣␣m e in V m V *, hence:
U jxVŽ.sÝxme␣␣me Ž.xgV,
U VŽ.xsÝ Žxne␣␣ .me, and ² U :²U: VVŽ.Ž.xsÝÝx,ee␣␣y e ␣␣,ex sxynx.
Ž. COROLLARY 4.3. If dim V k 1 mod p , V is a split injection.
PROPOSITION 4.4. Let : W ª V be an injection in CG. Assume that 22 2 dim W k 1Ž. mod p and that H : H W ª H V splits. Then splits. 502 SERRE AND FEIT
Proof. Consider the commutative diagram
6
WV 6 V
2VV* W6 Hm
6
Ј6 22 HWmW*HVmW*
2 where VWand are as above, Ј is equal to H m 1W* and is the 2 tensor product of the identity endomorphism of H V with the natural projection *: V * ª W *. By Corollary 4.3, applied to W, W is split; by assumption, Ј is split. Hence ( VW( s Ј( is split, and this implies that is split.
2 THEOREM 4.5. If H V is semisimple and dim V k 2Ž. mod p , then V is semisimple.
2 Proof. We have to show that every injection W ª V splits. Since H V 22 is semisimple, the injection H W ª H V splits. If dim W k 1Ž. mod p , Proposition 4.4 shows that W ª V splits. Assume now that dim W ' 1 Ž.mod p . Let W 0be the orthogonal complement of W in V *, i.e., the 0 kernel of the projection V * ª W *. We have dim W ' dim V y 1Ž. mod p , 0 2 hence dim W k 1Ž. mod p , since dim V k 2 Ž. mod p . By duality, H V *is 0 semisimple. The first part of the argument, applied to W ª V *, shows 0 that W ª V * splits, and hence W ª V splits. The next theorem describes the structure of V in the exceptional case left open by Theorem 4.5Ž. for explicit examples, see Sect. 7.3 :
2 THEOREM 4.6. Assume H V is semisimple and V is not. Then V can be decomposed in CG as a direct sum:
V s E [ W1[ иии [ Whh Ž.G0, Ž.) where:
ᎏthe Wii are simple, and dim W ' 0Ž. mod p ; ᎏE is a nonsplit extension of two simple modules W, WЈ such that dim W ' dim WЈ ' 1Ž. mod p .
ŽŽ.Note that ) implies dim V ' dim E ' 2 Ž. mod p , as in Theorem 4.5. . Proof. Using induction on the length of a Jordan᎐Holder¨ sequence of V, we may assume that V has no simple direct factor whose dimension is 0 Ž.mod p . GROUP REPRESENTATIONS 503
Let W be a simple subobject of V. Let us show that dim W ' 1Ž. mod p . If not, Proposition 4.4 would imply that W ª V splits, hence V s W [ VЈ 2 Ž for some VЈ g CG. Clearly VЈ is not semisimple but H VЈ is because it is 2 a subobject of H V .. By Theorem 4.5, applied to V and VЈ, we have dim V ' dim VЈ ' 2Ž. mod p ,
hence dim W ' 0Ž. mod p , which contradicts the hypothesis that V has no simple direct factor of dimension divisible by p. Hence, we have dim W ' 1Ž. mod p . Moreover, the injection W ª V does not split. Indeed, if V would decompose in W [ VЈ, we would have dim VЈ ' 1Ž. mod p , and Theorem 4.5, applied to VЈ, would show that VЈ is semisimple, hence also V, which is not true. The module W is the only
simple submodule of V. Indeed, if W1 were another one, the argument Ž. Ž . above would show that dim W11' 1 mod p , hence dim W q W ' 2 Ž. mod p since W l W11s 0. By Proposition 4.4, the injection W [ W ª V would split, and so would W ª V, contrary to what we have just seen. 2 Now put WЈ s VrW. We have dim WЈ ' 1Ž. mod p , and H WЈ is 2 semisimpleŽ because it is a quotient of H V .. By Theorem 4.5, WЈ is semisimple. At least one of the simple factors of WЈ has dimension k 0 Ž.mod p . Let S be such a factor, and let VS be its inverse image in V,so Ž. that we have W ; VSS; V. One has dim V k 1 mod p ; by Proposition 4.4, this shows that we may write V as a direct sum VS [ VЉ.IfVЉ/0, it contains a simple subobject, which is distinct from W, contrary to what was proved above. Hence we have VЉ s 0, i.e., S s WЈ, which shows that WЈ is simple and that V is a nonsplit extension of two simple objects W, WЈ with dim W ' dim WЈ ' 1Ž. mod p . There are similar results for Sym2 V. First:
2 THEOREM 4.7. If Sym V is semisimple and dim V k y2Ž. mod p , then V is semisimple.
2 ProofŽ. sketch . The argument is the same as for H V, using symmetric analogues VVVVand of and :
2 V :VªSym V m V *, 2 V : Sym V m V * ª V. Proposition 4.2 is replaced by
VV( s Ž.1 q n 1V where n s dim V. Ž. Hence V is a split injection if dim V k y1 mod p . Proposition 4.4 2 2 remains valid when H V is replaced by Sym V and 1 is replaced by y1. 504 SERRE AND FEIT
The same is true for the proof of Theorem 4.5Ž. with 2 replaced by y2, with one difference: 2 2 2 In the case of H we have used the fact that H V and H V * are dual to each other. The analogous statement for Sym2 V and Sym2 V * is true 2 when p / 2, but is not true in general for p s 2; the dual of Sym V is the space TS2 V * of symmetric 2-tensors on V *, which is not Sym2 V *. Fortu- nately, the case p s 2 does not give any trouble. Indeed: 2 PROPOSITION 4.8. If Sym V is semisimple and p s 2, then V is semi- simple.
2 F Proof. Let F: k ª k be the Frobenius map ¬ , and let V be the representation of G deduced from V by the base change F. The F-semi- 2 linear map V ª Sym V defined by x ¬ x и x gives a k-linear embedding of V F into Sym2 V, which fits into an exact sequence:
F 2 2 0 ª V ª Sym V ª H V ª 0. Since Sym2 V is assumed to be semisimple, so is V F. This means that V becomes semisimple after the base change F: k ª k. By an elementary resultŽwx 1, §13, no. 4, Proposition 4. this implies that V is semisimple. Remark. More generally, the same argument shows:
p Sym V semisimple « V semisimple if the characteristic p is ) 0. The analogue of Theorem 4.6 is:
THEOREM 4.9. If Sym2 V is semisimple and V is not, then V can be Ž. decomposed as V s E [ W1 [ иии [ Whh G0,where:
ᎏthe Wii are simple, and dim W ' 0Ž. mod p ; ᎏE is a nonsplit extension of two simple modules whose dimensions are congruent to y1Ž. mod p . The proof is the same.
COROLLARY 4.10. One has dim V G 2 p y 2. Indeed, it is clear that dim E G Ž.Ž.p y 1 q p y 1.
5. HIGHER EXTERIOR POWERS
m The results of Sect. 4 can be extended to H V for any m G 1Ž cf. Theorem 5.2.1 below. . We start with several lemmas. GROUP REPRESENTATIONS 505
5.1. Extension Classes Associated with an Exact Sequence Let
0 ª A ª V ª B ª 0Ž. 5.1.1 be an exact sequence in CG. We denote by Ž.V its class in the group
1 1 ExtŽ.B, A s H Ž.G, Hom kŽ.B, A s H ŽG, A m B*. .
A cocycle representing this class may be constructed as follows: select a Ž. k-linear splitting f: B ª V, and, for every s g G, define csf in Ž. Žy1.Ž. Hom k B, A as the map x ¬ s и fs xyfx, for x g B. Then cf is a 1-cocycle on G with values in Hom kŽ.B, A , which represents the class Ž.V . One has Ž.V s 0 if and only if f can be chosen to be G-linear, i.e., if and only if the injection A ª V splits.
m 5.1.2. The Filtration of H V Defined by A We view A as a subobject of V. For every integer ␣ with 0 F ␣ F m, let m F␣ be the subspace of H V generated by the x1 n иии n xmisuch that x ␣ belongs to A for i F ; put Fmq1 s 0. The F␣ are G-stable, and they m define a decreasing filtration of H V:
m иии H V s F01> F > Fmm> F q1s 0.
m One has Fm s H A. More generally, the quotient V␣␣␣s F rF q1can be ␣  identified with H A m H B, where  s m y ␣; in this identification, an Ž. element x1 n иии n xm of F␣ with xi g A for i F ␣, as above corre- sponds to
иии иии Ž.x1n n x␣␣m Ž.x q1n n xm, where xiiis the image of x in B. ␣ 2 Assume now G 1, and put V␣ s F␣y1rF␣q1. We have an exact se- quence
2 0 ª V␣ª V␣␣ª V y1ª 0,Ž. 5.1.3
Ž2.1Ž U. hence an extension class V␣ in HG,V␣␣mVy1. ␣  Ž2. Since V␣ s H A m H B, we may view V␣ as an element of the cohomology group
1 ␣  ␣ 1  1 H Ž.G, H A m H B m H y A* m H q B*.Ž. 5.1.4 506 SERRE AND FEIT
2 5.1.5. Comparison of the ClassesŽ. V and Ž V␣ . ␣1 The exterior product Ž.u, x ¬ u n x defines a map from H yA m A to ␣ H A, hence a CG-morphism: ␣y1 ␣ ␣ ␣y1 A,␣: A ª Hom kŽ.H A, H A s H A m H A*.Ž. 5.1.6 The same construction, applied to B* and to  q 1, gives q1  B*, q1: B* ª H B* m H B.Ž. 5.1.7  By tensoring these two maps, and multiplying by Ž.y1 , we get ␣  ␣y1 q1 ⌰␣: A m B* ª H A m H B m H A* m H B*.Ž. 5.1.8
Since ⌰␣ is a CG-morphism, it defines a map 1 1 1 ␣  ␣y1 q1 ⌰␣: H Ž.G, A m B* ª H Ž.G, H A m H B m H A* m H B*.
1 LEMMA 5.1.9. The image by ⌰␣ of the classŽ. V of Ž5.1.1 .is the class 2 ŽVof␣ .Ž5.1.3 . . ProofŽ. sketch . Select a k-splitting f ofŽ. 5.1.1 . Using f, one may identify the exterior algebra HV with H A m H B. This defines a k-split- 2 ting f␣ of V␣ . An explicit computationŽ. which we do not reproduce shows that the cocycle c corresponding to f is the image by ⌰ of the cocycle f␣ ␣␣ cf . Hence the lemma. 1 The next step is to give criteria for ⌰␣ to be injective. Put a s dim A and b s dim B,Ž. 5.1.10 so that we have dim V s a q b.Ž. 5.1.11
LEMMA 5.1.12. Assume Ž.ay1 0 Ž mod p .. Then the morphism ␣ y 1 k A, ␣ defined abo¨e is a split injection. ŽŽ.x Ž.Ž.. Recall thaty is the binomial coefficient xxy1иии x y y q 1 ry!
ProofŽ. sketch . Consider the CG-morphism ␣ ␣y1 A*, ␣ : A* ª H A* m H A, and let X ␣ ␣y1 A*, ␣ : H A m H A* ª A be its transpose. One has
X a y 1 A*, ␣ (A, ␣ s и 1A in EndŽ.A . Ž 5.1.13 . ž/␣y1 This identity is proved by a straightforward computation: one chooses a ␣ ␣ k-basis of the vector space A; this gives bases of H A, H A*, . . . ; one GROUP REPRESENTATIONS 507 determines the corresponding matrices, etc. The details are left to the reader. OnceŽ. 5.1.13 is checked, Lemma 5.1.12 is obvious.
EMMA Ž.b y 1 Ž . L 5.1.14. Assume  k 0 mod p . Then the morphism B*, q1 defined abo¨e is a split injection. Proof. This follows from the preceding lemma, with A replaced by B* and ␣ by  q 1. LEMMA 5.1.15. Assume a y 1 b y 1 и k 0Ž. mod p . ž/␣y1ž/ Then:
Ž.i The CG-morphism ⌰␣ defined in Ž5.1.8 .is a split injection. Ž.ii The map 1 1 1 ␣  ␣y1 q1 ⌰␣: H Ž.G, AmB* ªH Ž.G, H AmH BmH A*mH B* is injecti¨e. Proof. AssertionŽ. i follows from Lemmas 5.1.12 and 5.1.14 since the tensor product of two split injections is a split injection. AssertionŽ. ii follows from assertionŽ. i .
LEMMA 5.1.16. Assume a y 1 b y 1 и k 0Ž. mod p . ž/␣y1ž/ If the exact sequence Ž.5.1.3 splits, then A ª V is a split injection. Ž 2 .Ž2.Ž. Proof. We have V␣ s 0 by hypothesis. Since V␣ is the image of V 1 1 by ⌰␣ Ž.cf. Lemma 5.1.9 and ⌰␣ is injectiveŽ. cf. Lemma 5.1.15 , we have Ž.Vs0.
5.2. Semisimplicity Statements
Let V be as above an object of CG , and m an integer G 1. m THEOREM 5.2.1. Assume that H V is semisimple, and that the integer dim V has the following property:
Ž.) For e¨ery pair of integers a, b G 1 with a q b s dim V, there exists an integer ␣, with 1 F ␣ F m, such that a y 1 b 1 и y k 0Ž. mod p . Ž. 5.2.2 ž/␣y1ž/my␣ Then V is semisimple. 508 SERRE AND FEIT
Proof. Let A be a subobject of V, and let B s VrA. We want to show that A ª V splits. We may assume that A / 0, B / 0. Put as above a s dim A and b s dim B. m We have a, b G 1 and a q b s dim V. Choose ␣ as inŽ. 5.2.2 . Since H V is semisimple, the same is true for its subquotients, and in particular for 2 V␣ Ž.cf. Sect. 5.1.2 . Hence the exact sequenceŽ. 5.1.3 splits. By Lemma 5.1.16, this implies that A ª V splits. EXAMPLE 5.2.3. If m s 2, ␣ may take the values 1 and 2 andŽ. 5.2.2 means:
b y 1 k 0Ž. mod p if ␣ s 1, ay1k0Ž. mod p if ␣ s 2. If dim V s a q b is not congruent to 2Ž. mod p , one of these two is true. 2 Hence H V semisimple « V semisimple, and we recover Theorem 4.5. Here are two other examples:
3 THEOREM 5.2.4. Assume H V is semisimple and
dim V k 2, 3Ž. mod pifp/2, dim V k 2, 3Ž. mod 4 if p s 2. Then V is semisimple. Proof. Here ␣ may take the values 1, 2, 3 andŽ. 5.2.2 means
Ž.Ž.Žb y 1 b y 2 r2 k 0 mod p .if ␣ s 1, Ž.Ž.Ž.ay1by1k0 mod p if ␣ s 2, Ž.Ž.Žay1ay2r2k0 mod p .if ␣ s 3. If p / 2, these conditions mean, respectively,
b k 1, 2Ž. mod p , ak1Ž. mod p and b k 1 Ž. mod p , ak1, 2Ž. mod p .
If a q b k 2, 3Ž. mod p , it is clear that one of them is fulfilled. The case p s 2 is similar; the only difference is that the congruence
Ž.Ž.Ž.x y 1 x y 2 r2 k 0 mod 2 means that x k 1, 2Ž. mod 4 . GROUP REPRESENTATIONS 509
m THEOREM 5.2.5. Assume that H V is semisimple and dim V k 2,3,...,m Ž.mod p . Then V is semisimple.
Proof. Consider first the case p s 0Ž. see also Sect. 6.1 below . By assumption we have dim V / 2, 3, . . . , m.Ž Note that it is a priori obvious that these dimensions have to be excluded.. We may assume dim V / 0, 1, hence dim V ) m.If aqbsdim V, with a, b G 1, we put ␣ s 1 q supŽ. 0, m y b . We have a y 1 G ␣ y 1 and b y 1 G m y ␣ hence both Ž.a y 1 and Žb y 1 . are / 0. HenceŽ. 5.2.2 is satisfied. ␣ y 1 m y ␣ Suppose now p ) 0. The hypothesis dim V k 2, 3, . . . , m Ž.mod p implies pGm. Hence conditionŽ. 5.2.2 may be rewritten as
ak1,2,..., ␣y1Ž. mod p and bk1,2,...,my␣ Ž.Ž.mod p . 5.2.26 If b k 1, 2, . . . , m y 1Ž. mod p , we put ␣ s 1 andŽ. 5.2.6 holds. If b ' i Ž.mod p with 1 F i F m y 1, we put ␣ s m y i q 1. One has
a k 1,2,...,␣y1Ž. mod p , because otherwise dim V would be congruentŽ. mod p to i q 1,...,m, which would contradict our assumption. HenceŽ. 5.2.6 holds
5.3. Higher Symmetric Powers ␣ We assume here that p ) m Ž.or p s 0 , so that the dual of Sym V for ␣ ␣ F m is Sym V *. m THEOREM 5.3.1. If Sym V is semisimple and dim V k y2, y3,...,ym Ž.mod p , then V is semisimple. ProofŽ. sketch . One rewrites the previous sections with exterior powers replaced by symmetric powers. The sign problems disappear. Moreover, the integerŽ.Ža y 1 of 5.1.13 . becomes Ža q ␣ y 1 . . The rest of the proof is the ␣ y 1 ␣ y 1 same.
6. FURTHER REMARKS
6.1. Characteristic Zero
When p s 0, the theorems of Sects. 2᎐5 can be obtained more simply by the following methodŽ essentially due to Chevalleywx 2. : We want to prove that a linear representation V of G is semisimple, m knowingŽ. say that H V is semisimple and dim V / 2, 3, . . . , m. By enlarg- 510 SERRE AND FEIT ing k, we may assume it is algebraically closed; we may also assume that GªGLŽ.V is injective and that its image is Zariski-closed; hence G may be viewed as a linear algebraic group over k Žmore correctly: as the group of k-points of an algebraic linear group. . Seewx 6 for these easy reduction steps. Let U be the unipotent radical of G Žmaximal normal unipotent subgroup. . Because the characteristic is 0, an algebraic linear representa- m tion of G is semisimple if and only if its kernel contains U. Since H V is assumed to be semisimple, this shows that U is contained in the kernel of m GLŽ.V ª GL ŽH V .Ž. If dim V / 2, 3, . . . , m, this kernel is of order m if dim V ) m.Ž.or is a torus if dim V - 2 ; such a group has no nontrivial unipotent subgroup. Hence U s 1, and the given representation G ª GLŽ.V is semisimple.
6.2. Generalizations All the results of Sects. 2᎐5 extend to linear representations of Lie algebras, and also of restricted Lie algebras Ž.if p ) 0 . This is easy to check.
A less obvious generalization consists of replacing CG by a tensor category C o¨er k, in the sense of Delignewx 3 . Such a category is an abelian category, with the following extra structures: Ž. Ž . a for every V12, V g ob C ,a k-vector space structure on C Hom Ž.V12, V ; Ž. Ž. b an exact bifunctor C = C ª C, denoted by V12, V ¬ V 1m V 2; Ž. c a commutativity isomorphism V12m V ª V 21m V ; Ž. Ž. Ž. d an associativity isomorphism V12m V m V 3ª V 1m V 23m V .
These data have to fulfill several axioms mimicking what happens in CG Žcf.wx 3. . For instance, there should exist an object ‘‘1’’ with 1 m V s V for C every V, and EndŽ. 1 s k; there should be a ‘‘dual’’ V * with V ** s V and C C Hom Ž.W, V * s Hom Ž.V m W, 1 for every W g ob Ž.C ; etc. If V g obŽ.C , there are natural morphisms
1 ª V m V * and V m V * ª 1.
C The dimension of V is the element of k s EndŽ. 1 defined by the composition
1 ª V m V * ª 1.
It is not always an integer. All the results of Sects.2 and 3 are true for C pro¨ided the conditions on dim Vordim W are interpreted as taking place in k. For instance, if V m W is semisimple and dim W / 0inŽ.k, then V is semisimple. The proofs GROUP REPRESENTATIONS 511 require some minor changes: e.g., in Lemma 3.1, one needs to define directly the morphisms
fVV: V m V m V * ª V and j : V ª V m V m V *.
Moreover, the basic equality fVV( j s 1 V is one of the axioms of a tensor category,Ž cf.wx 3 ,Ž.. 2.1.2 . 2 2 As for the results of Sect. 4 on H V and Sym V, they remain true at least when p / 2, but some of the proofsŽ. e.g., that of Proposition 4.2 have to be written differently. I am not sure of what happens with Sect. 5: I have not managed to rewrite the proofs in tensor category style. Still, I feel m m that Theorem 5.2.5Ž on H V .Žand Theorem 5.3.1 on Sym V .should remain true whenever m!/ 0in kŽ.i.e., p s 0orp)m. Remark. An interesting feature of the tensor category point of view is the following principle, which was pointed out to me by Deligne: mm Any result on H implies a result for Sym , and con¨ersely Žhere again we assume m!/ 0in k.. This is done by associating to each tensor category C the category CЈ s superŽ.C , whose objects are the pairs Ž. VЈsV01,Vof objects of C; such a VЈ is viewed as a graded object VЈsV01[V, with grading group Zr2Z. The tensor structure of CЈ is defined in an obvious way, except that the commutativity isomorphism is modified according to the Koszul sign rule: the chosen isomorphism Ž.Ž.Ž.Ž. between 0, V11m 0, W and 0, W 11m 0, V is the opposite of the obvi- ous one. With this convention, one finds that
dim VЈ s dim V01y dim V .
In particular, if V g obŽ.C , one has dimŽ. 0, V sydim V. Moreover, one checks that
Ž.SymmV,0 if m is even, mŽ.0,V Hs ½Ž.0, SymmV if m is odd.
m Hence any general theorem on the functor H , when applied to CЈ, gives a corresponding theorem for the functor Symm, with a sign change in dimensionsŽ. compare for instance Theorem 4.5 and Theorem 4.7 .
7. EXAMPLES
The aim of this section is to construct examples showing that the congruence conditions on dim W and dim V in Theorems 2.4, 4.5, and 5.2.5 are best possible. We assume that p is ) 0 and that k is algebraically closed 512 SERRE AND FEIT
7.1. The Group G Let C be a cyclic group of order p, with generator s. Choose a finite abelian group A on which C acts. Assume
Ž.7.1.1 the order<< A of A is prime to p, and ) 1. Ž.7.1.2 the action of C on A y Ä41 is free.
Let G be the semidirect product A и C of C by A. It is a Frobenius group, with Frobenius kernel A. = Let X s HomŽ A, k . be the character group of A; we write X addi- x tively and, if a g A and x g X, the image of a by x is denoted by a . The group C acts on X by duality, and conditionŽ. 7.1.2 is equivalent to:
Ž.7.1.3 The action of C on X y Ä40 is free Ž.i.e., sx s x « x s 0.
x If x g X, denote by 1 g CA the k-vector space k on which A acts ¨ia Ž. Gx the character x. The induced module WxsIndAG 1 is an object of C , of dimension p. One checks easily:
Ž. Ž. Ž . 7.1.4 If x / 0, W x is simple and projecti¨e in CG .
Moreover
Ž. Ž. 7.1.5 E¨ery V g ob CG splits uniquely as V s E [ P, where E is the subspace of V fixed under A, and P is a direct sum of modules WŽ. x , with xgXyÄ40.
If we decompose V in V s [Vxx, where V is the A-eigenspace relative x to x Ži.e., the set of ¨ g V such that a и ¨ s a ¨ for every a g A., one has E s V0 and P s [y / 0 Vy. FromŽ. 7.1.5 follow:
Ž.7.1.6 V is semisimple if and only if the action of C on E is tri¨ial Ži.e., if and only if E ( 1 [ иии [ 1.. Ž.7.1.7 V is projecti¨e if and only if E is C-projecti¨e Ži.e., if and only if E is a multiple of the regular representation of C..
Note that bothŽ. 7.1.6 and Ž. 7.1.7 apply when E s 0, i.e., when no element of V, except 0, is fixed by A.
Ž. Ä4 Ž 7.1.8 Let x be an element of X y 0. If V s V0 i.e., if A acts trivially on V .Ž, then V m W x. is isomorphic to the direct sum of m copies of WxŽ.,where m s dim V. GROUP REPRESENTATIONS 513
Indeed, V is a successive extension of m copies of 1, hence V m WxŽ.is a successive extension of m copies of WxŽ.; these extensions split since WxŽ.is projective Ž 7.1.4 . ; hence the result.
7.2. Examples Relati¨e to Theorem 2.4 We reproduce here an example due to W. Feit, showing that the congruence condition ‘‘dim W k 0Ž. mod p ’’ of Theorem 2.4 is the best possible:
PROPOSITION 7.2.1. Let G be a finite group of the type in Sect. 7.1, and let n, m be two positi¨e integers, with m ) 1 and n di¨isible by p. There exist Ž. V,Wgob CG such that: Ž.i dim V s m and dim W s n; Ž.ii V is not semisimple; Ž.iii V m W is semisimple. Proof. Choose: V s a non-semisimple C-module of dimension m Žsuch a module exists since m ) 1;. Ž.иии Ž . Ä4 WsWx1 [ [ Wxnrpi, with x g X y 0. The projection G ª C makes V into a G-module with trivial A-action. It is clear thatŽ. i and Ž ii . are true. ByŽ. 7.1.8 , V m W is isomorphic to the direct sum of m copies of W, hence it is semisimple.
7.3. Examples relati¨e to Theorems 4.5 and 5.2.5 The following proposition shows that the congruence conditions of Theorem 5.2.5 are the best possible: PROPOSITION 7.3.1. Let i and n be two integers with 2 F i F p, n ) 0 and n ' i Ž.mod p . There exists a finite group G of the type described in Sect.
7.1 and an object V of CG such that: Ž.a dim V s n; Ž.b V is not semisimple; m Ž.c H V is semisimple for e¨ery m such that i F m F p. The case i s 2 gives the following result, which shows that the condition dim V k 2Ž. mod p of Theorem 4.5 is the best possible: COROLLARY 7.3.2. If n ) 0 and n ' 2Ž. mod p , there exist a finite group 2 G and a non-semisimple G-module V such that dim V s n and H Vis semisimple. m ŽEven better: H V is semisimple for 2 F m F p.. 514 SERRE AND FEIT
Proof of Proposition 7.3.1. We need to choose a suitable G s A и C of the type described in Sect. 7.1. To do so, write n as n s i q hp, with h G 0. LEMMA 7.3.3. There exist a finite abelian group X, on which C acts, and h elements x1,..., xofXh ,with the following properties: Ž.7.3.4 The action of C on X y Ä40 is free. Ž. Ž. 7.3.5 For e¨ery family I1,...,Ih of nonempty subsets of wx0, p y 1 the relation
h j ÝÝsx␣s0 Ž.) ␣s1jgI␣ implies I␣s wx0, p y 1 for ␣ s 1,...,h. Proof. Assume first that h s 1. In that case,Ž. 7.3.5 just means that, if I << Ý j is a subset ofwx 0, p y 1 , with 0 - I - p, one has jg I sx1 /0. This is easy to achieve: choose some integer e ) 1, prime to p, and define X1 to be the augmentation module of the group ring ZreZwxC , i.e., the kernel of Ž. ZreZwxCªZreZ; put x11s 1 y s. It is easy to check that X , x1has the required property. If h ) 1, one takes for X the direct sum of h copies of the C-module
X11defined above, and one defines x ,..., xhto be
Ž.Ž.x11,0,...,0 ,..., 0,...,0, x .
Proof of Proposition 7.3.1 Ž.Žcontinued . Let X, x1,..., xh .be as in Lemma 7.3.3. PropertyŽ. 7.3.4 implies
< = hence < V s E [ WxŽ.1[иии [ Wx Ž.h,Ž. 7.3.6 where E is a non-semisimple C-module of dimension i Žviewed as a .Ž.Gx␣Ž. G-module with trivial action of A , and Wx␣ sIndC 1 cf. Sect. 7.1 . We have dim V s i q hp s n, and it is clear that V is not semisimple. It m m remains to check that H V is semisimple if i F m F p. ByŽ. 7.3.6 , H V is a direct sum of modules of type: a b1 bh H E m H WxŽ.1mиии m H WxŽ.Ž.h, 7.3.7 GROUP REPRESENTATIONS 515 with a q b1 q иии qbh s m. Let us show that every such module is a semisimple. If all b␣ are 0, we have a s m G i, and H E is 0 if m ) i and a HEs1if msi. HenceŽ. 7.3.7 is either 0 or 1 and is semisimple. We may thus assume that one of the b␣ is ) 0. By using induction on h,we may even assume that all the b␣ are ) 0. Since b1 q иии qbh s m y a F p, the b␣ are F p. Suppose one of them, say b1, is equal to p. We have then b2 s иии s bh s 0, m s p, a s 0, Ž. pŽ. and the G-module 7.3.7 is equal to H Wx1 s1, hence is semisimple. We may thus assume that 0 - b␣ - p for every ␣. Observe now that, if xgXyÄ40 , the characters of A occurring in the A-module WxŽ.are x, sx,...,spy1x, and their multiplicity is equal to 1. Hence the characters b Ž. Ý j occurring in H Wx are of the form jg I sx, for a subset I ofwx 0, p y 1 Ž. with < h ÝÝsxj , ␣s1jgI␣ Ž. with < APPENDIX by Walter Feit A.1 Let G be a finite group, let p be a prime and let k be a field of 2 characteristic p ) 0. If V is a kGwx-module of dimension n such that H V is semisimple, then V is semisimple unless n ' 2Ž. mod p by Theorem 4.5. Similarly, Theorem 4.6 asserts that if Sym2 Ž.V is semisimple then so is V unless n ' y2Ž. mod p . Corollary 7.3.2 implies that, for odd p, Theorem 4.5 is the best possible result. In Theorem A2 below, it is shown that for p s 2, for infinitely many but not all even n, there exists a non-semisimple module V of dimension n 2 with H V semisimple. 516 SERRE AND FEIT Furthermore, Theorem A1Ž. ii shows that if p is a prime such that m p<Ž2 q 1. for some natural number m, then there exist infinitely many integers n and non-semisimple modules V of dimension n with Sym2 V semisimple. If p is odd there always exist infinitely many natural numbers m such m that p<Ž2 y 1.Ž . The situation is more complicated in case ii. of Theorem m A1. By quadratic reciprocity p<Ž2 q 1.Ž for some m if p ' 3 or 5 mod 8. , m and p ¦ Ž2 q 1.Ž for any m if p ' 7 mod 8.Ž . In case p ' 1 mod 8. , such an m may or may not exist, the smallest value in this case where no such m exists is p s 73. It is not known whether a non-semisimple V exists with Sym2 V semisimple in case p is a prime such as 7, 23, . . . which does not m divide 2 q 1 for any natural number m. m If p s 2 q 1 is a Fermat prime, Corollary 4.10 implies that the module Vconstructed in Theorem A1Ž. ii has the smallest possible dimension m 1 2 q s 2 p y 2. For no other primes is it known to us whether there exists a non-semisimple module V with Sym2 V semisimple of the smallest possible dimension 2 p y 2. The method of proof of Theorem A1 also yields some additional 2 examples for all odd primes, of non-semisimple modules V with H V semisimple. Basic results from modular representation theory are used freely below. See, e.g.,wx 4 . The following notation is used, where p is a prime and G is a finite group: F is a finite extension of ޑp, the p-adic numbers; R is the ring of integers in F; is a prime element in R. From now on it will be assumed that k s Rr R is the residue class field. Moreover both F and k are splitting fields of G in all cases that arise. PROPOSITION A.1.1. Let ␣ s ␣12q ␣ be a Brauer character of G, where ␣1 is the sum of irreducible Brauer characters which are afforded by projecti¨e modules, and ␣2 is the sum of irreducible Brauer characters i, no two of which are in the same p-block. Then any kwx G -module W which affords ␣ is semisimple. Proof. Since a projective submodule of any module is a direct sum- mand, W s W12[ W , where Wiiaffords ␣ . Furthermore, W1is the direct sum of irreducible projective modules, and so is semisimple. W2 is semisimple as all the constituents of an indecomposable module lie in the same block. GROUP REPRESENTATIONS 517 An immediate consequence of Proposition A.1.1 is COROLLARY A.1.1. Let s ⌺ijq ⌺ be a character of G, where each ijand is irreducible. Let U be an R-free Rwx G -module which affords . Suppose that the following hold: Ž.i i has defect 0 for each i. Ž.ii j is irreducible as a Brauer character for each j. Ž.iii If j / jЈ then jЈ and j are in distinct blocks. Then U s UrU is semisimple. Corollary A.1.1 yields a criterion to determine when a module is semisimple, which depends only on the computation of ordinary charac- ters. To construct a non-semisimple kGwx-module the following result is helpful. PROPOSITION A.1.2Ž Thompson, seewx 4, I.17.12. . Let U be a projecti¨e indecomposable Rwx G -module. Let V be an F wx G -module which is a sum- mand of F m U. Then there exists an R-free Rwx G -module W with F m W f V and W s WrW indecomposable. COROLLARY A.1.2. Let s q be the character afforded by a projec- ti¨e indecomposable Rwx G -module, where and are characters. Then there exists an indecomposable Rwx G -module which affords as a Brauer character. A.2 Seewx 5, pp. 355᎐357 for the results below. Let D denote a dihedral group of order 8 and let Q be a quaternion group of order 8. Let m be a natural number. Up to isomorphism there are two extra-spe- 2 m 1 cial groups of order 2q , TmŽ., for s "1. The first, TmŽ., 1 , is the central product of m copies of D, while TmŽ.,y1 is the central product of Qwith m y 1 copies of D. Let T s TmŽ.,, let Z be the center of T and 2 let T s TrZ. The map q s qmŽ.,:TªZ with qyŽ.sy defines a nondegenerate quadratic form on T, where Z is identified with the field of two elements. TmŽ., 1 has a maximal isotropic subspace of dimension m, while TmŽ.,y1 has a maximal isotropic subspace of dimension m y 1. Furthermore the orthogonal group Oq2 mŽ.,acts as a group of automor- phisms of TmŽ.,. Ž. m Oq2 m ,has a cyclic subgroup of order 2 y which acts regularly on m TmŽ.,. Thus if p is a prime with p<Ž.2 y , then T has an automor- phism of order p whose fixed point set is Z. Let P s ²: and let G be the semidirect product PT. Then G s PT is a Frobenius group with Frobenius kernel T. 518 SERRE AND FEIT Every irreducible character of G which does not have T in its kernel is induced from a nonprincipal linear character of T and so has degree p. T has one faithful irreducible character . Thus extends to an irreducible character of G in p distinct ways. The values of are easily computed. Hence all characters of G can be described as follows: Ž. PROPOSITION A.2.1. i G has p linear characters 1 s 1,...,p, whose kernels contain T. Ž.ii The irreducible characters of G which do not ha¨e T in their kernel all ha¨e degree p, and so are of p-defect 0. Ž.iii There is a faithful irreducible character ˜ of G such that ˜˜1,...,pare all the faithful irreducible characters of G. Furthermore, m ˜˜Ž.1s2 and ¨anishes on all elements of T y Z. Ž.iv There are two p-blocks of G of positi¨e defect. The sets of irre- Ä 4Ä 4 ducible characters in them are ii<1 F i F p and ˜ <1 F i F p , respec- ti¨ely. Ž.v E¨ery irreducible character of G is irreducible as a Brauer charac- ter. The notation of this subsection, especially of Proposition A.2.1, will be used freely below. A.3 THEOREM A1. Assume that p / 2. m Ž.i Let m be a natural number such that p<Ž2 y 1.. Then there exists a finite group G and a non-semisimple kwx G -module V of dimension 2 mq1' 2 2 Ž.mod p such that H V is semisimple. m Ž.ii If p<Ž2 q 1.for a natural number m, then there exists a finite m 1 group G and a non-semisimple kwx G -module V of dimension 2 q' y2 Ž.mod p such that Sym2V is semisimple. Proof. Let G be as in the previous subsection. As a Brauer character, G ˜˜iis ޑ-valued for all i. As the induced character s Ý i,itisthe character afforded by a projective indecomposable kGwx-module. Hence by Corollary A.1.2 there exists an indecomposable kGwx-module V which affords the Brauer character s 2 ˜˜˜, since q iagrees with 2 ˜as a Brauer character. Since is irreducible, 2 restricted to T contains the principal character of T with multiplicity 4. Thus 2 is the sum of four linear Brauer characters and irreducible Brauer characters of p-defect 0. Vis a kGwx-module, and hence also a kTwx-module. As T is a pЈ-group, Brauer characters of T are ordinary characters. Let skew be the character GROUP REPRESENTATIONS 519 2 of T afforded by H V and let sym denote the character of T afforded by Sym2 V. Then for y in T 2 2 skewŽ.y s Ž. Ž.y y Ž.y r2, 22 symŽ.y s Ž. Ž.y q Ž.y r2. Let s "1 denote the Frobenius᎐Schur index of . Ý Ž 2 . << Ý Ž2. << Then y g Tyy s T . Hence gTy s 2 T . Therefore 1 ÝskewŽ.y s 2 y , < In particular, both skew and sym contain the principal character as a constituent. Therefore the Brauer characters sy, sk of G afforded by Sym2 V and by 2 H V, respectively, both contain at least one linear constituent. Hence each contains at most three linear constituents as V m V has exactly four linear constituents. m 1 m 1 Ž.isk1 Ž .s Ž22q Ž q y1..r2, hence skŽ.Ž. 1 ' 1 mod p . As p G 3 this implies that sk s 1 q , where  is the sum of irreducible 2 projective Brauer characters. Thus H V is semisimple. m 1 m 1 Ž.ii sy Ž. 1 s Ž22q Ž q q1..r2, hence syŽ.Ž. 1 ' 1 mod p . As p G 3 this implies that sy s 1 q , where  is the sum of irreducible projective Brauer characters. Thus Sym2 V is semisimple. Remark. The argument in the proof of Theorem A1 involving the Frobenius᎐Schur index is only needed for p s 3. If p ) 3, then the last two statements in the proof are clear. Serre has pointed out that V can be defined directly as E m X, where E is an indecomposable two-dimensional module of P s GrT and X is the irreducible G-module afforded by ˜ as a Brauer character. A.4 THEOREM A2. Suppose that p s 2. Let q ' 3Ž. mod 8 be a prime power and let G s SLŽ. 2, q . Then there exists a non-semisimple kwx G -module V of 2 dimensionŽ. q q 1 r2 such that H V is semisimple. Proof. The irreducible characters in the principal 2-block of PSLŽ. 2, q Ž. Ž . are 1, St, 12, and , where i1 s q y 1 r2 for i s 1, 2. The restriction 520 SERRE AND FEIT of i to a Borel subgroup is irreducible and so i is irreducible as a Brauer character. The remaining 2-blocks of PSLŽ. 2, q are either of defect 0or1. Ž. Ž. There are q y 3 r8 2-blocks Bi of PSL 2, q of defect 1. The Brauer tree of each Bi has two vertices and one edge, and so every irreducible character in Bi is irreducible as a Brauer character. Let i1 and i2 be the irreducible characters in Bi. The notation can be Ž. Ž. chosen so that i1 u s 2 and i2 u sy2 for an involution u. SLŽ. 2, q has a faithful irreducible character of degree Ž.q q 1 r2 Ž. whose values lie in ޑ 'y q with s 1 q 1 as a Brauer character. By Corollary A.1.2 there exists an R-free RGwx-module W which affords such that V s WrW is indecomposable. The center of G acts trivially on 22 HWand so H W is an RwPSLŽ. 2, q x-module. Direct computation shows 2 2 that H W affords 1 q ⌺i1. By Corollary A.1.1, H V is semisimple. REFERENCES 1. N. Bourbaki, Modules et anneaux semi-simples, in ‘‘Algebre,’’` Chap. VIII, Hermann, Paris, 1958. 2. C. Chevalley, ‘‘Theorie´ des Groupes de Lie,’’ Vol. III, Hermann, Paris, 1954. 3. P. Deligne, Categories´ tannakiennes, in ‘‘The Grothendieck Festschrift,’’ Vol. II, pp. 111᎐195, Birkhauser,¨ Boston, 1990. 4. W Feit, ‘‘The Representation Theory of Finite Groups,’’ North-Holland, Amsterdam, 1982. 5. B. Huppert, ‘‘Endliche Gruppen’’ I, Springer-Verlag, BerlinrNew York, 1967. 6. J.-P. Serre, Sur la semi-simplicite´´ des produits tensoriels de representations de groupes, In¨ent. Math. 116 Ž.1994 , 513᎐530.