Representation Theory of Finite Groups: Proceedings of a Special Research Quarter at the Ohio State University, Spring 1995

= ljf(g)f(a) and f(acjJ(g» = f(a)ljf(g) Vg

E

G, a E A.

A block b of a finite group G is a primitive idempotent of the center of the group algebra <9G; the algebra <9Gb is called the block algebra of b.

94

Radha Kessar

Let H be a subgroup of G. The Brauer map of OG with respect to H, denoted Br H is the natural map from (OG) H onto the quotient (OG)H I

L

Tr~ ((OG)R)

+ J(O)(OG)H.

RcH

Here Tr~ ((OG)R) is the ideal in (OG)H which consists of all elements of the type LXER/ H x a, where a is an element of (OG)R. A defect group P of b is maximal among the subgroups of G such that Br p (b) is not zero. A source algebra of b is an algebra of the form WGbi, where i is a primitive idempotent of the algebra (OGb)P for which Brp(i) is not zero. A source algebra iOGbi of b is naturally an interior P-algebra via the homomorphism which maps an element x of P to the element ix of WGi. Actually, the notion of source algebra is not restricted to block algebras of finite groups but exists for any interior G-algebra [PI]. In some sense, the block algebra OGb can be " almost" computed from the knowledge of its source algebra as an interior P -algebra. More precisely, we have the following theorem which is due to Puig. Theorem 1.1 ([PI], [P2]). 1.

The algebras OGb and iOGbi are Morita equivalent, i.e. they have equivalent module categories.

2.

The matrix of generalized decomposition numbers of the block b of G, its local category, the vertices and sources of its indecomposable modules, are all determined by the isomorphism class (as interior P -algebra) of the source algebra iOGbi of b.

In the following example, we explicitly describe the source algebras for certain blocks. This example is taken from [Br]. Example 1.2. Let r be a prime number such that p divides r - 1, (an) be an infinite sequence of integers none of which is divisible by m be an integer such that m < p. We put G n = Glm(r an ) and we by B n the subgroup of G n consisting of the upper triangular matrices We put

and let p. Let denote in G n .

Let P be a Sylow p subgroup of G n . Then P is isomorphic to a direct pr oduct of m copies of the Sylow p subgroup of ('Ill r'll) x. Also, in is a

On Blocks and Source Algebras

95

primitive idempotent of (OGn)p and the algebra inOGi n is a source algebra for the principal block of G n . On the other hand, inOGi n is isomorphic to the algebra O(P x Nc, (P)/CCI (P)).

2. Puig's Conjecture One of the main problems of block theory is to classify the block invariants given the isomorphism type of its defect groups[A]. As is seen from Theorem 1.1 above, the classification of source algebras, given the defect groups of the corresponding blocks, would be quite a step forward in this effort. The following conjecture was announced by Lluis Puig at the Group Theory Conference held at Oberwolfach in 1982. Let P denote a fixed non-identity p group. Conjecture 2.1 (Puig). There are only finitely many isomorphism classes of interior P -a1gebras which are source algebras for blocks of finite groups having P as a defect group. The above conjecture has been verified in its entirety for the family of cyclic p-groups. This result is due to Markus Lincklemann; in fact he gives necessary

and sufficient conditions for two blocks with a common cyclic defect group to have isomorphic source algebras [L]. In a slightly different vein, the conjecture has been verified in some cases if instead of considering blocks of any finite group, only blocks of certain infinite families of groups are considered. For example, if we restrict ourselves to blocks of p-solvable groups, then Puig's conjecture is known to hold; the proof uses the classification of the finite simple groups [H].

2.2 The groups Sn. It is well known [JJ that the ordinary irreducible characters of the symmetric group Sn on n letters are in one to one correspondence with the set of partitions of n and that two irreducible characters of X and rJ of Sn belong to the same p block of Sn if and only if the partitions to which they correspond have the same p-cores. Further, the Murnaghan-Nakayama formula calculates the restriction of an irreducible character X of Sn to a subgroup Srn in terms of removals of hooks from the corresponding partitions. Using this machinery, Joanna Scopes [Se] found a suitable criterion which, if satisfied by two p-blocks , b of Sn and c of Srn, would force the block algebras kSnb and kSrnc to be Morita equivalent. The transitive extension of this criterion subdivides the blocks of the symmetric groups into finitely many families; as a consequence, she proved that Donovan's Conjecture [AJ holds for the blocks of the symmetric groups, namely that there are only finitely

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many Morita equivalence classes of blocks of the symmetric groups that have P as a defect group. Later, Puig showed that the equivalence demonstrated by Scopes could actually be obtained through an isomorphism of the corresponding source algebras [P31 He proved the following theorem.

Theorem 2.2.1. Let band c be blocks of Sn and Sm respectively with defect group P. If band c satisfy Scopes criterion of equivalence then there exists an isomorphism (1.1)

as interior Sm x Sn-m algebras. Further, this isomorphism induces an interior P -algebra isomorphism between a source algebra of b and a source algebra of c.

Corollary 2.2.2. Puig's conjecture holds for the blocks of the symmetric groups. Though the results of Scopes and Puig are essentially the same, the techniques they employ for the proof are quite different. Scopes' proof relies on the bridge between both the ordinary and the modular character theory of the symmetric groups and the theory of partitions whereas in Puig's proof only the correspondence at the level of ordinary irreducible characters is used. It would seem therefore that Puig's method would lend itself more easily to generalization. In fact, Puig believes that his method will yield similar comparisons between blocks in any family of finite groups where the ordinary irreducible characters allow a parameterization similar to that of the symmetric groups. Let us try to outline Puig's general idea. For a finite group G and a block b of G, let Irr( G, b) denote the set of ordinary irreducible characters of G that belong to G. For the purpose of later applying this method to other families of groups, let us use G n to denote Sn. We note that if m < n, Gm may naturally be considered a subgroup of G n and we assume this inclusion. The proof of Theorem 2.2.1 is roughly as follows. I

Given a block of b of G n for n sufficiently large, the group Gm (with C of it are chosen such that there is an integer rand there is a 1-1 correspondence (given by Xn --* Xm) between Irr( G n , b) and Irr( Gm, c) such that for any element Xn of Irr(G n, b), the restriction of Xn to the block c of Gm is equal to r copies of Xm' This easily yields the rank comparison m < n) and a block

rankCJ(cb()Gnbc) = r 2 rankCJ(()Gmc).

On Blocks and Source Algebras

11

97

Since Brp(b) and Brp(c) are respectively blocks of kCGn(P)jZ(P) and kCG m(P)jZ(P)

of defect zero, Brp(c) Brp(b)kCGn(P)jZ(P) Brp(c) and Br p(c)kCG n (P)jZ(P) Br p(c) are simple algebras, and it is easy to analyze the structure of the fixed point algebra (Brp(c) Brp(b)kCGn(P)jZ(P) Brp(c))NGmCP). This information is then "lifted" to give information about the idempotents of the fixed point algebra (cbr:JGnc)G m. It turns out that there is exactly one conjugacy class of primitive idempotents in (cbr:JGnc)G m whose elements are not in the kernel of the Brauer map with respect to P. III Let S be a unitary r:J simple subalgebra of (cbr:JGnc)G m. Then we have a decomposition (cbr:JGnc) :::: S 0CJ CcbCJGnc(S).

The rank comparison in Step I along with a lemma ofPuig [PS, Proposition 3.8] is then used to show that r:JGmc and CcbCJGnc(S) are isomorphic as interior Gm algebras, thus yielding the result. This approach, if modified slightly, can be applied to the covering groups of Sn as we describe below. 2.3 The groups Sn and An. A presentation of the group Sn via generators and relations can be found in [HH]. We have an exact sequence: -

B

1 ---+ (z) ---+ Sn ---+ Sn ---+ 1 where z is a central involution of Sn. The group Sn is a covering group for Sn for n > 3. Let An denote the inverse image, 0- 1 (An), of the alternating group An under 0, An is a double cover of An. Let us assume from now on that p is odd. Let e : r:JS n ---+ r:JS n denote the natural extension of 0 to the group algebra of Sn. Then the image e(b) of a block b of Sn (respectively An) is either 0 in which case b is called a faithful block of Sn (respectively An ) or (b) is a block of Sn (respectively An) in which case b is called a non-faithful block of Sn (respectively An). Also, if b is a faithful block of Sn with a non-trivial defect group, then b is also a block of r:JAn, under the natural inclusion of r:JAn in r:JSn . The faithful ordinary irreducible characters (irreducible characters belonging to faithful blocks) of Sn are parameterized by the set of strict partitions of

e

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Radha Kessar

n; a strict partition A of n indexes a single character X).. of

Sn

if A is even and A indexes a pair of associate characters X).. and xf if A is odd [8]. If two characters X and 17 of Sn are in the same block of Sn, then the partitions by which they are indexed have the same p -bar core. Conversely, if the partitions indexing X and 17 have the same p-bar core, then either X and 17 are in the same p-block or X and 17 are associate characters and each of them belongs to a block of defect zero [Cl, [H], [0], A. O. Morris [Ml] [M2] has given a formula which relates the restriction of characters of Sn to Srn to the removal of bars from the corresponding partitions. Thus the situation for the double covers is quite similar to that of the symmetric groups. We indicate briefly how Puig's method for proving Theorem 2.2.1 must be modified for these groups. Let the group G n denote now Sn. Since the correspondence between the strict partitions of n and the ordinary faithful irreducible characters of n fails to distinguish between a character and its associate, a correct correspondence between the characters of G n in band the characters of c is obtained only through a finite sequence bl, bz ... b t of blocks (each with P as a defect group) of intermediate groups G nj with n > ni > ni+1 > m for 1 ::: i < t. In other words, in Step I we obtain a rank comparison between rank o (cab<9G nbac) and the algebras ranko (<9G rn c) , where a is the product of the bi 'so The analysis in Step II yields that there are two conjugacy classes of primitive idempotents in (cab<9G n ac)G m whose elements are not in the kernel of the Brauer map with respect to P; hence in step III we work with the algebra I cab<9G n acI where I is an idempotent, suitably chosen, of (cab<9G n ac)G m • In order to get the required result we need also to consider the action of An on (cab<9G nac)G m. This actually points to an analogous result for the groups An, and in fact we obtain the following theorem. Let 'B p denote the set of faithful blocks in the family of groups {Sn k:~o which have a defect group isomorphic to P. For b E 'B p, let Ibl = n if b is a block of Sn.

Theorem 2.3.1. 1.

There exists an integer N p with the following property: For any b in 'B p with n = jbl > N p there exists c in 'B p with m = lel < Ibl such that there is an idempotent I of (<9S n b)Sm for which

On Blocks and Source Algebras

99

as interior Sm aLgebras. In particuLar; there is an interior P -aLgebra isomorphism between a source aLgebra of b (considered as a bLock of Sn) and a source aLgebra of c (considered as a bLock of Sm ). The statem:nt of (I) remains valid on repLacing the group Sn by An and the group Sm by Am.

2.

As an immediate corollary of the above theorem and of Corollary 2.2.2 we get

Corollary 2.3.2. Puig's conjecture hoLds for the p-bLocks of the groups Sn, for every odd prime p. Corollary 2.3.3. Puig's conjecture hoLdsfor the faithfuL p -bLocks ofthe groups An, for every odd prime p. Remark 2.3.4. A result of Puig and Lincklemann on central p extensions proves Corollary 2.3.2 for the case p = 2 as a consequence of Corollary 2.2.2 Remark 2.3.5. For n > 3 and n i= 6, there is another central extension of Sn, usually denoted by Sn. However, the character theory of Sn may be described in exactly the same fashion as that of Sn [C, Section 3.9] and it is easy to see_that th~ results described in this section remain valid if we replace the group Sn by Sn.

References [A]

J. L. Alperin, Local representation theory, in: The Santa Cruz Conference on Finite Groups (B. Cooperstein and G. Mason, eds.), Proc. Symp. Pure Math. 37, Amer. Math. Soc., Providence, R.L, 1980, 369-375.

[B]

M. BroUl~, Theorie locale des blocs, in: Proceedings of the International Congress of Mathematicians, Berkeley, 1986, 360-368.

[C]

M. Cabanes, Local structure of the p-blocks of Sn, Math Z. 198 (1988), 519-543.

[H] [HH]

[J]

J. F. Humphreys, Blocks of projective representations ofthe symmetric groups, J. London Math. Soc. (2) 33 (1986), 441-452. P. N. Hoffman and J. F. Humphreys, Projective Representations of the Symmetric Groups, Oxford Mathematical Monographs, Oxford University Press, New York, 1992. G. D. James, The Representation Theory of the Symmetric Groups, Lecture Notes in Math. 682, Springer-Verlag, Berlin 1978.

100

Radha Kessar

[K]

Radha Kessar, Blocks and source algebras for the double covers of the symmetric and alternating groups, preprint, 1995.

[L]

Markus Lincklemann, The isomorphism problem for cyclic blocks and their source algebras, preprint, 1994.

[M 1]

A. O. Morris, The spin representation of the finite group, Proc. London Math. Soc. (3) 12 (1962), 55-76.

[M2]

A. O. Morris, On Q-functions,1. London Math. Soc. 37 (1962),445-455.

[0]

J. Olsson, On the p-blocks of the symmetric and alternating groups and their covering groups, 1. Algebra 128 (1989),188-213.

[P I]

L. Puig, Pointed groups and construction of characters, Math Z. 176 (1981), 358-369.

[P2]

L. Puig, Local fusions in block source algebras, 1. Algebra (1986),358-369.

[P3]

L. Puig, On Joanna Scopes' criterion of equivalence for blocks of symmetric groups, Algebra Colloq. I: 1 (1994),25-55.

[S]

I. Schur, Uber die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitionen, J. Reine Angew. Math. 139 (1911), 155-250.

[Sc]

1. Scopes, Cartan matrices and Morita equivalence for blocks of the symmetric groups, J. Algebra 142 (1991), 441-455.

Department of Mathematics Yale University New Haven eT 06520 U.S.A.

A Survey on the Local Structure of Morita and Rickard Equivalences between Brauer Blocks Lluis Puig

1. Introduction 1.1. Since Jeremy Rickard's thesis - developing a "Morita theory" for the equivalences between the so-called derived categories of the categories of modules over two algebras [15], and exhibiting such an equivalence when the starting two algebras are the block algebras in characteristic p of Brauer blocks with the same cyclic defect gorups and the same inertial quotients - it has appeared an ample belief in the sense that the existence of such equivalences could "explain" the similarities between some pairs of Brauer blocks which, however, are far from being Morita equivalent. Of course, the first example is Rickard's equivalence for blocks with cyclic defect groups mentioned above, lifted by Markus Linckelmann to characteristic zero [6] and explicited by Raphael Rouquier which exhibits in [21] a quite simple two terms bicomplex inducing such an equivalence. Although, at present, there are not so many other examples, the reasonably expected derived equivalences exposed by Michel Broll(~ in [3], [4] and [5] justify, from our point of view, an effort to understand a priori the consequences of such equivalences between the socalled local structures [2], [1] and [8] of the concerned blocks, in particular seeking inductive constructions. A first attempt in that direction is Rickard's result on the so-called splendid equivalences in [18], which we improve here (Section 6). 1.2. But before trying to understand the relationship between the local structures of two blocks having equivalent derived categories of the categories of modules, it is prudent to start by analysing this relationship when they have already equivalent module categories; that is to say, in a widely employed terminology, when the blocks are Morita equivalent. Let us fix some notation; as usual, p is a prime number and eJ is a complete discrete valuation ring having an algebraically closed residue field k = (') / J (eJ) of characteristic p (we allow the possibility eJ = k), and all the modules we consider are finitely

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L1uis Puig

generated over <9. Let G and G' be finite groups, band b' be respectively blocks of G and G' (i.e. primitive idempotents in Z(<9Gb) and Z(<9G'b'» and finally Py and p y be respectively maximal local pointed groups on A = <9Gb and A' = <9G'b' (i.e. P and P' are respectively defect groups of band b', whereas y and y' are respectively conjugacy classes of primitive idempotents i in A P and i' in (A')P! such that i fJ. A~ and i' fJ. (A')~: for any proper subgroup Q of P and Q' of P'). We denote respectively by AO = <9Gb o and A'o = <9G'b'o the opposite interior G-algebras of A and A', and respectively by Ay and A~! the source algebras of band b'; that is to say, Ay is the <9-algebra i Ai , where i E y, endowed with the group homomorphism P -+ (i Ai)* mapping u E P on ui (see [10], §2 for more detail on the notation). !

1.3. First of all, let recall the current cases we know where band b' are Morita equivalent; they are so in the following three situations:

I.

The blocks band b' are nilpotent and the defect groups P and p' are isomorphic ([9], Main theorem).

2.

The groups G and G' are p-solvable and the groups obtained by iterating Fong's reduction are isomorphic ([11]).

3.

The group G is a Chevalley group over a finite field of characteristic different from p and, for some parabolic subgroup H of G, some block e of H such that eb = e , some Levi complement L in H of the radical of H and some block f of L such that f e = e, formed by cuspidal irreducible characters, we have G' = NG(L, 1), b' = f, P' = P C L ([12], Corollary 5.10).

In all these situations, it happens that P and P' are isomorphic, so that we may assume that P = P' , and that there is an indecomposable <9 P -module N of vertex P such that, setting 5 = Endo (N) and considering it as an interior P -algebra, we have an interior P -algebra embedding Ay -----* 5 ®o A~!

(1.3.4)

(i. e. injective homomorphism with image i"(5 ®o A~/)i" where i" is the image of the unity element) and P stabilizes an <9-basis of 5. Actually, it is not difficult to prove that, conversely, the existence of such embedding forces the symmetric one

A> -----* 50 ®o Ay and implies that band b' are indeed Morita equivalent.

(1.3.5)

Morita and Rickard Equivalences

103

1.4. On the other hand, according to a well-known definition, band b' are Morita equivalent if (and only if) there is an O-free O(G x G')-module M" associated with b 0 (b')O such that, denoting by (M")* the dual O-module Hom(') (M", 0) which is an O-free O(G x G')-module too, associated with bO 0 (b'), we have repectively O(G x G) and O(G' x G')-module isomorphisms (1.4.1) and our first purpose had been to connect these ismorphisms with those embeddings. A first remark in that direction is that these module isomorphisms can be easily modified to obtain algebra isomorphisms; indeed, recall that isomorphisms 1.4.1 imply that, in particular, the restriction of M" to O(l x G') is projective, so that we get an O(G x G)-module isomorphism (1.4.2) but, it is easily checked that this isomorphism ought to be the structural interior G-algebra homomorphism (1.4.3) multiplied by some element in End(') (M")GxG' , which forces homomorphism 1.4.3 to be bijective. 1.5. Another easy consequence of isomorphisms, 1.4.1 is that M" is an indecomposable O( G x G') -module; hence, it has a vertex P" and an 0 P" -source N", and if we are interested in relating to each other the local structures of A and A', it is reasonable to employ the local structure of M" to get it. So, set S" = End(') (N,,) considered as an interior P" -algebra and recall that the induced interior G x G' -algebra Ind~,~G' (S") ([10],2.14) is just

End(')(Ind~,~G'(N")) in that case; now, since M" is a direct summand of the induced O(G x G')-module Ind~,~G' (N") , we get from 1.4.3 an interior G-algebra embedding (1.5.1) and to "compute" the second term, we have been led to introduce the noninjective induction of interior H -algebras.

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2. The Noninjective Induction 2.1. Let Hand H' be finite groups,

(2.1.1)

which is clearly compatible with the action of K by conjugation on B and on C @(lK B; moreover, if a, a' E Band K fixes 1 Q9 a in C Q9(lK B then, for any x E K we get (l Q9 a) . (x . a') = 1 Q9 aa'

(2.1.2)

consequently, the map 2.1.1 restricted to (CQ9(lK B)K induces a new K -stable bilinear map (C Q9(lK B)K

X

(C Q9(lK B) ---+ C Q9(lK B

(2.1.3)

which determines a product in (C Q9(lK B)K (C Q9(lK B)K x (C Q9(lK B)K ---+ (C @(lK B)K

(2.1.4)

mapping (1 @ a, 1 Q9 a') on 1 Q9 aa' for any a, a' E B such that K fixes 1 @ a and 1 Q9 a' in C Q9(lK B. It is easily checked that (C Q9(lK B)K with this product becomes an associative C-algebra and that the structural map H ---+ B* induces a group homomorphism K H = H / K ---+ (C Q9(lK B) .

(2.1.5)

2.2. We are ready to define the induced interior H' -algebra Indrp (B); considering (C @(lK B)K and CH' as C(H x H)-modules, the second via
(x' @ (l Q9 a) Q9 s')(y' @ (l @ a') @ t') =

{

(l Q9 a . Z . a') @ t'

~r

(2.2.2)

according to whether or not there is z E H such that
(2.2.3)

Morita and Rickard Equivalences

105

-1

mapping x' E H' on L y' x' y' 0 (l 0 1 B) 0 y' where y' runs over a set ofrepresentatives for H' j
x' 0 a . (z . n)

I

(x 0(l0a)0s ' )·(y'0n)=

I

~r

(2.3.1)

according to whether or not there is Z E H such that
2.3.2. If N is 0 -free and B ~ End o (N) as interior H -algebra then the action of Indrp(B) on Indrp(N) induces an interior H' -algebra isomorphism Indrp(Endo(N))

~

Endo(Indrp(N)).

2.4. Let HI! be a third finite group and
o 0 o[

(0 00K B)K ----+ 0 00L B

(2.4.1)

which is clearly compatible with the action of L by conjugation and therefore it induces a canonical map (2.4.2) which, unfortunately, needs not to be bijective. It can be proved that this map induces an interior HI! -algebra homomorphism Indrp/(Indrp(B)) ----+ Indrp/orp(B)

(2.4.3)

which is bijective if and only if the map 2.4.2 is so. In particular, this happens when one of the group homomorphisms

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by left and right multiplication, is a direct summand of a permutation (J(L x L)-module.

2.5. Let B ' be an interior H' -algebra; now, we will compare the interior It is not difficult to H'-algebras Indcp(B) 00 B ' and Indcp(B 00 Rescp(B ' see that there is a unique interior H' -algebra homomorphism

».

(2.5.1)

»

mapping (l 0 (l 0 a) 0 I) 0 a ' on I 0 (l 0 (a 0 a' 0 I for any a E B such that K fixes I 0 a in (J 00 K B, and any a ' E B ' ; it makes sense since K fixes 10 (a 0a ' ) in (J 00K (B 00 Rescp(B ' Moreover, if B ' is (J-free we get an interior H' -algebra isomorphism

».

(2.5.2) 2.6. The corresponding version of the so caled "Mackey formula" needs here a pull-back of groups as follows. Let L' be a third finite group and r/ : L' ~ H' a second group homomorphism, and consider the following pull-back diagram of groups

cp H

~

H'

(2.6.1)

L

~

L'

It is clear that, up to suitable identification, cp and 1/1 have the same kernel K; from this fact it is quite clear that we have a canonical interior L' -algebra embedding

(2.6.2) which is an isomorphism if and only if H' = cp(H) . r/(L ' ). More generally, if we consider all the pull-backs determined by the family {(1J')X' }x" where x' runs on a set of representatives for 1J1(L I )\H' jcp(H) in H', then the very Mackey formula states that the image of the unity elements of the corresponding interior L ' -algebra embeddings form a pairwise orthogonal idempotent decomposition of the unity element in Indcp(B).

Morita and Rickard Equivalences

107

2.7. There is no difficulty on inducing any homomorphism between two interior H -algebras to obtain an interior H' -algebra homomorphism between the corresponding induced interior H' -algebras, and on proving that Indip becomes a functor between the categories of interior H - and H' -algebras. Then, all the homomorphisms above are in fact natural maps between suitable functors and, to be complete, it has to be checked all the obvious compatibilities between them, which amounts to prove the commutavity of certain diagrams. 2.8.

Finally, let us consider the relationship between the Brauer section B(Q) of B at any p-subgroup Q of H (i.e. the interior CH(Q)-algebra

k Q9CJ (BQ / B~) where R runs on the set of proper subgroups of Q) and the corresponding Brauer section (Indip(B))(ep(Q)) of Indip(B). It seems hopeless to obtain a precise general answer but if we assume that ep is SUfjective and that, considering the action of H x H on B by left and right muliplication and denoting by L\ (H)) the diagonal subgroup of H x H, B is a direct summand of a permutation <:J«K x K)· L\(H))-module with projective restriction to <:J(K x K) we get the following result:

LR

2.8.1. If Q' is a p-subgroup of H' then the natural homomorphism B K (Q/) ---+ (<:J 0CJK B)K (Q/) determines, for any complement Q of K in ep-l(Q/) an interior ep(CH(Q))-algebra embedding eip.Q(B) : IndipQ(B(Q)) ---+ (Indip(B)(Q/) where epQ : CH (Q) ---+ ep(CH(Q)) is the restriction of ep, and the family {eip. Q (B) (l 0 B rQ (l B))} QEC, where C is a set of representatives for the set of K -orbits on the set of K -complements of ep-l (Q/), is apairwise orthogonal idempotent decomposition of the unity element of (Indip(B))(Q/), which does not depend on the choice of C.

This result depends on the following lemma which generalizes Lemma 4.2 in [18].

Lemma 2.9. Let X be a finite group, Y a normal subgroup of X and M a direct summand of a permutation <:JX -module with projective restriction to <:JY. Then for any p-subgroup Q' of X' = X/Y, the inclusion MY C M induces a kNxf(Q/)-module isomorphism (MY)(Q/) ~

(n

M(Q»)Y

(2.9.1)

Q

where Q runs on the set of complements of Y in the inverse image of Q' in X.

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L1uis Puig

3. Morita Equivalences between Brauer Blocks 3.1. Let us come back to the end of Section 1; keeping all the notation there, recall that we need to compute the interior G-algebra (Ind~,~G' (S"»I xG'. Denote by Jr : G x G' --+ G and Jr' : G x G' --+ G' the projection maps and set A" = Ind~,~G' (S"); applying successively 2.4.3, 2.5.2 and 2.6.2 (in particularly simple situations ), we have the interior G-algebra isomorphisms

Res~,~G' (A") ~ Ind7l' (Indg:?' (Resg:f (A"»)

~ Ind7l' (A" ®c Indg~f(0» ~ Ind7l' (A" ®c Res7l',(Indf (0»)

(3.1.1)

and the composed isomorphim maps a" E A" on 1 ®Tr:~?' (a" ®(l ® 1® 1»; inparticular,itmaps a" E (A")lxG' on 1®(a"®Tr:~?'(l®I®I», so that it induces an interior G-algebra isomorphism (A,,)lxG' ~ Ind7l' (A" ®c Res7l',(OG'»

(3.1.2)

Hence, denoting respectively by p : p" --+ G and p' : p" --+ G' the restrictions of Jr and Jr', and applying again 2.5.2 and 2.4.3, we get finally

(Ind~,~G' (S,,»lxG' ~ Ind7l'(Ind~,;G'(S") ®c Res7l',(OG'»

~ Ind7l'(Ind~,;G' (S" ®c Resp,(OG'») ~ Indp(S" ®c Resp,(OG'»

(3.1.3)

and setting Ker(p) = 1 X V' for a suitable p-subgroup V' of G', for any family {s;, }X'EG' of elements of S" such that 1 x V' fixes LX'EG's;, ® x' in 0 ®c(1 xV') (S" ®c Resp,(OG'», the composed isomorphism h fulfills

h(Tr~~~:(

L x'EG'

X,-l

® s;, ® I» = 1 ® (l ® (

L

s;/ ® x'» ® 1 (3.1.4)

x'EG'

We are ready to state our first result; to formulate it as a necessary and sufficient condition, we keep just our notation in 1.2.

Theorem 3.2. The blocks band b' are Morita equivalent if and only if, for a suitable p-subgroup P" of G X G' such that Jr(P") = P and Jr'(P") = P' and a suitable indecomposable 0 P" -module N" of vertex P" such that the restriction of N" to P" n (G xl) and to P" n (I x G') are both projective, we have an interior P -algebra embedding

Ay --+ Inda(S" ®c Resa'(A~,)

(3.2.1)

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109

where a : pll ---+ P and a' : pll ---+ p' are respectively the restrictions of nand n', and S" = End o (Nil). Remark 3.3. In that case, the Morita equivalence is realised by an indecomposable direct summand M" of Ind~;GI (Nil), which can be determined by a suitable multiplicity module ([10], 6.4 and [13], Theorem 3.4), from the multiplicity module of Py on A. Indeed, notice that embedding 3.2.1 determines a local point y of P on Inda(S"00Resal(A~I» and this interior P-algebra is obviously embedded in Inda(S" 00 Respl«(')G'» ;:: (Ind~,~G' (S"» , xG'

(3.3.1)

so in (Res~~g;(A"»'xGI; hence, y is also a point of P x G' on A" and the multiplicity module of Py on A is canonically isomorphic, always via embedding 3.2.1, to an indecomposable projective direct summand of the multiplicity module of (P x G')y on A" which, by a result of Laurence Barker ([1], Theorem 5.1), determines a unique point a" of G x G' on

A" ;:: Endo(Ind~,~GI (Nil»

(3.3.2)

so an indecomposable direct summand of Ind~,~GI(Nil) 3.4. It is now evident that embedding 3.2.1 becomes embedding 1.3.4 whenever a and a' are group isomorphisms, but it may be not yet clear to the reader why S" should have a P -stable (') -basis. It turns out that this is indeed a strong consequence of embedding 1.3.4 itself; precisely, we have the following general result which, as a matter of fact, has been a surprise for us. Recall that the so called local category [8] of the block b is the category where the objects are the local pointed groups on A = (,)Gb and the morphisms are the group exomorphisms induced by composing the suitable restrictions of inner automorphisms of G and the inclusions of pointed groups ([8], Definition 2.1) and that A determines a central k* -extension of the group of automorphisms of any object ([ 10], 6.6). Theorem 3.5. Let Q be a finite p-group, Rand R' interior Q-algebras and N an indecomposable (') Q -module. Assume that Rand R' have Q x Q' -stable (') -bases by left and right multiplication, that R(R) and R' (R) are nonzero symmetric k -algebras for any subgroup R of Q and that, setting S = End o (N), we have an interior Q-algebra embedding

R ---+ S

00

R'.

Then S has a Q -stable (') -basis by conjugation.

(3.5.1)

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Corollary 3.6. With the notation of Theorem 3.2, assume that band b' are Morita equivalent. Then the following conditions are equivalent to each other. 3.6.1. The homomorphism a : p" 3.6.2. The homomorphism a ' : p"

~ ~

P is bijective. pi is bijective.

3.6.3. The prime p does not divide rank o(N"). 3.6.4. The interior p" -algebra S" has a p" -stable ()-basis by conjugation. Moreover, in that case, embedding 3.2.1 induces an equivalence between the local categories of band b' which preserves the corresponding central k* -extensions.

3.7. When band b' are Morha equivalent and all the conditions in Corollary 3.6 hold, we say that the Morita equivalence is basic; in that case, it is now clear that, up to suitable identifications, embedding 3.2.1 becomes indeed embedding 1.3.4; moreover, notice that if N" ~ () then Ay ~ A>, a fact proved independently by Leonard Scott about 1990 [22]. It is remarkable that, when () has characteristic zero, all the known Morita equivalences between Brauer blocks are basic; actually, it can be proved from Theorem 3.2 that if () has characteristic zero and the blocks band b' are nilpotent then all the Morita equivalences between them are basic (Markus Linckelmann proves a stronger result in [7]), and a recent result of Klaus Roggenkamp and Leonard Scott [20] suggests that the same could be true for any pair of blocks whenever G and G' are p-solvable. A stronger consequence of Theorem 3.2 answers in the affirmative the question we raised in [9], 1.8. Theorem 3.8. Assume that () has characteristic zero. If band b' are Morita equivalent and b' is a ni/potent block then b is a ni/potent block too and the defect groups P and pi are isomorphic. 3.9. Actually, the results of this section can be generalized to the so-called Morita stably equivalences between Brauer blocks. We say that band b' are Morita stably equivalent if in 1.4 we have just stable isomorphisms or equivalently, if isomorphisms 1.4.1 are replaced by the ()(G x G)- and ()(G ' x G')-module isomorphisms M" 0oG" (M")* ~ A EB C and (M")* 00G M" ~ A' EB C' (3.9.1)

where C and C' are respectively suitable proective ()(G x G)- and ()(G ' x G ' ) -modules. Then Corollary 3.6 and Theorem 3.8 remain true when replacing "Morita equivalent" by "Morha stably equivalent", and "local category" by a suitable "stable local category" where we exclude the trivial p-subgroup; in

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111

particular, notice that, since Jeremy Rickard has proved in [16], 5.5 (see also [4], Proposition 5.2) that if band b' are derived equivalent then they are also Morita stably equivalent, if <9 has characteristic zero, band b' are derived equivalent and b' is nilpotent then b is nilpotent too and P ;::; pi.

4. The Differential Structures 4.1. Let H be a finite group; from now on, we have to replace <9 H -modules by complexes of <9 H -modules; fortunately enough, as we will see in the next section, we need only to consider complexes of <9H -modules which are still finitely generated over <9. Recall that a (finitely generated) complex of <9H -modules is just an <9H -module M (always finitely generated over (9) endowed with a gradation, which is nothing but a pairwise orthogonal idempotent decomposition {i~ }zEZ of idM in End(,)H (M), and a differential, which isjust an element d M of End(,)H(M) fulfilling (d M )2=0 and dM(i~(M))Ci~l(M)foranYZEZ

(4.1.1)

notice that, since M is finitely generated, all but a finite set of idempotents i~ are zero, and that the sum

L <9i~ + L <9i: d ZEZ

M

(4.1.2)

ZEZ

is an <9-subalgebra of End(,)H(M).

4.2. In other words, denoting by F the <9-valued functions over Z, by s h : F -----+ F the <9 -algebra automorphism fulfilling (sh(f))(z) = f(z

+ 1) for any Z E

Z and any f E F

(4.2.1)

and by D the <9-algebra containing F and an element d fulfilling d 2 = 0, df = sh(f)d for any f E F, and D = F Efl Fd,

(4.2.2)

M is actually a DH -module; that is to say, an <9-module endowed with a unitary <9 -algebra homomorphism DH -----+ End(') (M).

(4.2.3)

Coherently, we replace interior H -algebras (called also interior <9H -algebras in the sequel) by interior DB -algebras which, mimicking the case of End(') (M), are <9-algebras B endowed with a unitary <9-algebra homomorphism DB -----+ B

(4.2.4)

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As for interior <9 H -algebras, an homomorphism between two interior D H algebras is simultaneously a nonnecessarily unitary <9-algebra homomorphism and a D H -bimodule homomorphism.

4.3.

In order to define the tensor product between either DH -modules or interior D H -algebras, we have to consider the following <9 -subalgebra Do. Let Fo be the <9-subalgebra of F formed by the periodic functions; notice that Fo @o Fo can identified (and we do!) with the <9-algebra of all the <9-valued biperiodic functions over Z x Z (on the contrary, F @o F maps not surjectively into the <9-algebra of all the <9-valued functions over Z x Z); it is quite clear that (4.3.1)

Do = Fo EEl Fod

is an <9-subalgebra of D and the point is that Do admits a Hopf <9-algebra structure, namely with the diagonal <9-algebra homomorphism ~o

: Do ---+ Do @o Do

(4.3.2)

defined by

+ Z') for any z, Z' d @s+I@d

~o(f)(z, Z') = fez

and

~o(d)

=

E Z and any f E Fo,

(4.3.3)

where s E Fo maps z on (-If. Now, for any <9-algebra B (always finitely generated as <9-module!), and any <9-algebra homomorphism g : D ---+ B, it is quite clear that g(Do) = g(D) and it can be proved that for any <9-algebra homomorphism f : D @o D ---+ B, the composed homomorphism 60

Do ---+ Do

@o

Do C D

@o

f

D ---+ B

(4.3.4)

extends to a unique <9-algebra homomorphism ~~(f)

: D ---+ B

(4.3.5)

4.4. Consequently, if Band B ' are interior D H -algebras and we denote by sts : DH ---+ Band sts' : DH ---+ B ' the structural maps, the tensor product of Band B ' is the <9-algebra B @o B ' endowed with the <9-algebra homomorphism DH ---+ B

@o

B'

(4.4.1)

mapping x E H on sts(x)@sts'(x) and g E D on ~~0SI(sts@stsl)(g). When B = Endo(M) and B' = Endo(M'), we get the tensor product of D H -modules; more generally, it is easily checked that if L is a normal

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113

subgroup of H then the kernel of the canonical map

M Q9C) M' ---+ M Q9C)L M'

(4.4.2)

*

where the right en-module structure of M is given by y m = y-l . m for any y ELand any m EM, is a D H -submodule of M Q9C) M' and therefore M 0C)L M' becomes a D(HjL)-module. The associativity of the tensor product comes from the coassociativity of Doo and the uniqueness of the extension 4.3.5; similarly, the commutativity "up to isomorphism" comes from the equality so

0

Doo = int(to)

0

Doo

(4.4.3)

where So is the automorphism of Do Q9C) Do which exchanges both factors, to E Fo Q9C) Fo maps (z, z') on (-1)zz' and int(to) denotes the corresponding inner automorphism of Do 0C) Do. Moreover we consider the tJ-algebra isomorphism (4.4.4) defined by t(d) = sd and (t(f))(z) which fulfills over Do

=

f( -z) for any z E Z and any f E F

(4.4.5) and allow us to define the opposite interior D H -algebra BO of an interior DH-algebra B and,consequently,the tJ-dual M* of an tJ-free DH-module M. In particular, notice that an interior D H -algebra B has a canonical D(H x H)-module structure obtained from the tJ-algebra homomorphism B Q90 BO ---+ EndC)(B)

(4.4.6)

defined by left and right multiplication. 4.5. Let B be an interior D H -algebra; as when working with D H -modules, we have to consider the so-called cycles and bords of B; actually, for our purpose here, we can restrict ourself to consider only a-cycles and a-bords. It is not difficult to check from the D-module structure of B that the set Co(B) of a-cycles of B is the interior tJ H -subalgebra formed by the elements of B which centralize the image of D by the structural map stB : DH ---+ Band the set Bo(B) of a-bords of B is the ideal of Co(B) formed by the elements d . a + a . d where a runs on the set of elements of degree one of B; as usual, we set

Ho(B) = Co(B)j Bo(B) which is an interior (') H -algebra.

(4.5.1)

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L1uis Puig

4.6. More generally, if L is a subgroup of H then B L is an interior DCH(L)algebra and we consider also Co(B L )

= CO(B)L, Bo(B L ) C

Bo(B)L and Ho(B L )

= Co(BL)j Bo(B L ).

(4.6.1) Now a point E of L on B is a conjugacy class of primitive idempotents in Co(B L ) = CO(B)L, so it is just an ordinary point of L on the interior (') H -algebra Co(B) ([ 10],2.10) and we say that E is contractile if E C Bo(B); notice that if j E E then j Bj endowed with the map DL -+ j Bj defined by the multiplication by j is an interior D L -algebra, noted BE and obviously we have ([10], 2.13) (4.6.2) As in the interior (')H -algebra case, if B = Endo (M) then BE = End o (j (M)) and j (M) is an indecomposable directsummand ofthe DL-module Resf (M). Coherently, pointed groups on B are nothing but ordinary pointed groups on Co(B) and we define inclusion and localness from those on Co(B) ([10], 2.10); except that there are the contractile pointed groups which we are not interested in. Notice that a pointed group on B contained in a contractile one is itself contractile and it can be proved that a pointed group on B is contractile if and only if its defect pointed groups are so. 4.7. However, localness on interior DH -algebras is not so plain; indeed, if Q is a p-subgroup of H, it is clear that the Brauer section B(Q) of B at Q inherits an interior DCH(Q)-algebra structure from B, so that we can consider the interior (')CH(Q)-algebra Co(B(Q)). On the other hand, according to our definition, the set of local points of Q on B corresponds bijectively with the set of points of the Brauer section (Co(B))(Q) of Co(B) at Q ([10], 2.10.1). It is easily checked that we have a canonical interior CH (Q) -algebra homomorphism (Co(B))(Q) -+ Co(B(Q))

(4.7.1)

but it needs not be either injective or surjective; in particular, a local point of Q on B needs not to determine a point of 1 on B(Q). 4.8. There is a particular situation, which occurs in Rickard equivalence between blocks (see Section 5 below), where homomorphism 4.7.1 is an isomorphism. First of all, recall that a D H -module M is contractile if an only if idM belongs to Bo(Endo(M)) or equivalently, all the points of H on Endo(M) are contractile. On the other hand, any interior H -algebra can be considered

Morita and Rickard Equivalences

lIS

as an interior DH -algebra via the canonical (J-algebra homomorphism (4.8.1) mapping f E F on f (0) and d on zero; in particular, any (J H -module can be considered as a DH -module and we call it a u-restricted DH -module. Now, we call the direct sums of contractile and u -restricted D H -modules a-split D H -modules and it is not difficult to prove that if B is a a-split D Q-module then we have (Co(B))(Q) ~ Co(B(Q))

(4.8.2)

and for any subgroup L of NH(Q) containing Q such that Br~(BL) = B( Q)L, we have also

(4.8.3)

4.9. Finally, it remains to introduce the induction of interior D H -algebras. Let H' be a second finite group,

(4.9.1) and we extend to D H' the structural (J -algebra homomorphism (J H' ----+ Indrp(B) by mapping g E Don Tr:('H)O@00stB(g))01) Itisnotdifficultto check that homomorphisms 2.3., 2.4.1, 2.5.1 and 2.6.2 are interior D-algebra homomorphisms too when we consider them for interior D-algebras. Furthermore, if f3 is a point of H on B and LE a pointed group on B such that

(4.9.2) it is well known ([10], 2.14) that there is a unique exoembedding (Le. a unique conjugacy class of embeddings) of interior (JH -algebras

(4.9.3)

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L1uis Puig

such that the following diagram of canonical exoembeddings is commutative Resf (h~)

(4.9.4)

r Co(B)e

It can be proved that ii~ extends to a unique exoembedding of interior D Halgebras Bf3 ~ Indf (BE)

(4.9.5)

such that the corresponding extended diagram is commutative too.

s. Rickard Equivalences between Brauer Blocks 5.1. Let us come back to our standard notation in 1.2 and consider respectively A and A' as interior DG- and DG' -algebras, via the structural maps induced by homomorphism 4.8.1; in particular, they become respectively D( G x G)and D(G' x G')-modules 4.4.6. In [16], Jeremy Rickard proves that the socalled derived categories of the categories of A - and A' -modules are equivalent, as triangulated categories, to each other if and only if : 5.1.1. There is an indecomposable D(G x G')-module M" associated with b 0 (b')o such that its restrictions to (9(G x 1) and to (9(1 x G') are both projective and we have respectively D(G x G)- and D(G' x G')-module isomorphisms M" 0CJG ' (M")* ~ A EB C and (M")* 0CJG M" ~ A' EB C'

for suitable contractile modules Cover D( G x G) and C' over D( G' x G').

But notice that Rickard's result does not guarantee that the starting equivalence between those derived categories is induced by M". 5.2. As Rickard points out, in that case M" induces actually an equivalence between the so called homotopic quotients of the categories of DGb- and DG'b' -modules (i.e. the quotients by the homomorphisms which factorize throughout contractile modules) that we denote respectively by ModDGb and ModDGlb " and following Michel Broue [4], we call any equivalence between

Morita and Rickard Equivalences

117

ModDGb and ModDG'b' induced by an indecomposable D(G x G')-module as in condition 5.1.1 Rickard-equivalence between the blocks band b'. 5.3. So, let M" be an indecomposable D(G x G')-module, associated with b 0 (b')o, such that its restriction to tJ(G x 1) and to tJ(l x G') are both projective; it is clear that Endo(M,,)lxG' is an interior DG-algebra and, consequently, that Ho(End o (M")l xG') is an interior G-algebra (cf. 4.5.1) and Endo(M,,)lxG' has a canonical D(G x G')-module structure (cf. 4.4.6). Then, as in 1.4, a first remark is that M" induces a Rickard equivalence between band b' (i.e. we have the isomorphism in 5.1.1) if and only if End o (M")l xG' is a O-split D(G x G)-module and we have an interior tJG-algebra isomorphism A ~ Ho(Endo(M")lxG')

(5.3.1)

moreover, it can be proved that in that case, there is a bijection between the set of pointed groups HfJ on A and the set of noncontractile pointed groups H ~ on End o (M") I X G' such that H fJ and H ~ correspond to each other if and only if isomorphism 5.3.1 induces an interior tJH -algebra isomorphism AfJ ~ Ho«EndO<M")lxG')~)

(5.3.2)

5.4. Since M" is indecomposable, idMII is a primitive idempotent in Co(End o (M")) and thus a" = {idM II } is the unique point of G x G' on Endo(M"); let P;II be a defect pointed group (GxG')a" (either on Endo(M") as interior D(G x G')-algebra or on Co(Endo(M")) as interior tJ(G x G')-algebra); we still call P" a vertex of M" and any j" E y" determines an indecomposable direct summand N" = j" (M") of Res~,~ G' (M"), still called a D P" -source of M". Hence S" = End o (N") is an interior D P" -algebra and we consider respectively the interior D( G x G') - and DG -algebras A" = Ind~,~G' (S") and

A=

(A,,)lxG'.

(5.4.1)

By 4.9.5 we have a canonical exoembedding End o (M") ---+ A"

(5.4.2)

so that a" can be identified with a point of G x G' on A", but notice that, at the same time, it can be considered as a point of G on A; coherently, we identify respectively Endo(M"), S" and Endo(M")lxG' with A~", A~II and

Aall.

The point is that all interior G -algebra isomorphisms we state in

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L1uis Puig

3.1 to compute A are now compatible with the D-structures, so that, mutatis mutandis, we get the same interior DG-algebra isomorphism (5.4.3) where c:JG ' has the interior DG-algebra structure induced by homomorphism 4.8.1. We are ready to state the generalization of Theorem 3.2; as there, we keep just our notation in 1.2.

Theorem 5.5. The blocks band b' are Rickard equivalent if and only if there are a p-subgroup p" of G X G' such that n(P") = P and n'(P") = pi, an indecomposable D p" -module N" of vertex P" such that its restriction to c:J(P" n (G xl)) and to c:J(P" n (l x G ' )) are both projective and, setting S" = End(') (N") and denoting by (J : P" ~ P and (J' : P" ~ pi the respective retrictions of n and ni, a local point y of P on Inda(S" 0(') Resa,(A~,)) such that Inda(S" 0(') Resa,(A~,))y is a O-split D(P x P)-module and we have an interior P -algebra isomorphism (5.5.1)

Remark 5.6. Actually, as in Remark 3.2, there is a more precise result involving multiplicity modules which gives a necessary and sufficient condition on M" for inducing a Rickard equivalence between band b'. 5.7. From now on, we assume that M" induces a Rickard equivalence between b and hi; then, in 5.4 we may choose P;, in such a way that:

5.7.1. We have n(P") = P, n'(P") = pi and an interior DP-algebra embedding -yl/,y' . ~

hy

. Ay

~

"

I

Ind a (S 0(') Resu,(A y '))

where y is the noncontractile point of P on A such that Ay ~ Ho(A y ) (cf. 5.3.2). In order to give a more precise description of the relationship between the local categories of band b' (cf. 3.4), it is proving useful to consider the following definition (which has not been introduced in Section 3 to avoid too much redundancy). A triple (Qo, Q~I/' Q~,) of local pointed groups Qo on A, Q~" on A" and Q~, on A' is called a local tracing triple on A, A" and A' if we have n (Q") = Q, n ' (Q") = Q' and a canonical interior DQ-algebra exoembedding (cf. Remark 5.2 below) -8" 8'

hg '

~

: Ag

~

"

I

Indr(A o" 0(') Resr,(A lI ,))

(5.7.2)

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119

where r : Q" ----+ Q and r' : Q" ----+ Q' are respectively the restrictions of rr and rr', and 3 is the noncontractile point of Q on A such that A,s ~ Ho(Ag) (cf. 5.3.2). Notice that ( Py, P:", P~I ) is indeed a local tracing triple on A, A" and A', since A~II ~ 5". Recall that 3 is also a point of Q x G' on A", so on Co(A"), and let us denote respectively by VCO(AII)g(Q~II) and VA'(Q~,) simple modules over Co(A")9" ,s and (A')Q' where 8" and 8' act nontrivially. Theorem 5.8. A triple (Q,s, Q~II' Q~,) of local pointed groups Q,s on A, Q~" on A" and Q~, on A' is a local tracing triple on A, A" and A' if and only if Q~" is a defect pointed group of (Q x G')g on A", we have rr' (Q") = Q' and, considering respectively the structural maps g' and g" from OCC,(Q') to (A')Q' and to Co(A")j", Resgl(VA'(Q~,)) is a quotient of Resg,,(VCo(A")g (Q~II))' Remark 5.9. Notice that neither exoembedding 5.7.2 nor Theorem 5.8 are symmetric in the roles of A and A'; indeed, if ( Q,s, Q~II' Q~,) is a local tracing triple on A, A" and A' then ( Q~/' Q~II' Q(j ) needs not to be a local tracing triple on A', (A")o and A (i.e. with repect to the Rickard equivalence between b' and b induced by (M")*). However, if (Q~" R~II' RE) is a local tracing triple on A', (A")O and A, it can be proved that there is x E G such that (Q(j)X eRE' 5.10. It is clear that G x G' acts by conjugation on the set of local tracing triples on A, A" and A'. Moreover, if ( Q(j, Q~", Q~, ) and ( RE, R;/I' R~, ) are local tracing triples on A, A" and A', we say that (RE' R~II' R~, ) is contained in (Q,s, Q~", Q~,) if we have RE C Q,s on A, R~" C Q~II on A", R~, C Q~I on A' and, for a suitable canonical exoembedding (cf. Remark 5.12 below) of interior D R -algebras, the following diagram is commutative ResW(hr",sI)

(5.10.1)

h:''',f' f

Ag

Indv(A~1I 00 Resv,(A~/))

where v : RI! ----+ R and v' : R" ----+ R' are respectively the restrictions of rr and rr'. Then it is easily proved that the inclusion between local tracing

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L1uis Puig

triples is transitive and we consider the category where the objects are the local tracing triples on A, A" and A', and the morphisms are the group exomorphisms between the middle terms induced by composing the suitable restrictions of inner automorphisms of G x G' and the inclusions of local tracing triples: we call it the local category of the Rickard equivalence induced by M".

Theorem 5.11. The functor from the local category of the Rickard equivalence induced by M" to the local category of the block b defined by the first projection is full and induces a bijection between the sets of isomorphism classes of objects in both categories.

Remark 5.12. Actually, to give a precise definition and to prove the unique-8" 8'

8" 8'

ness of the canonical exoembeddings h.' in 5.7.2 and gf ,,',f in 5.10.1, we 8 need to introduce the Higman envelope of an interior D H -algebra B, namely the smallest interior DH -algebra G(B) admitting interior DH -algebra exoembeddings I

(f(B) : B ---+ G(B) and hI(! : lnd: (BI(!) ---+ G(B)

(5.12.1)

for any local pointed group TI(! on B, such that the following evident diagram of exoembeddings is commutative ([ 10], 2.13.1 and 2.14.1) Res: ((f(B)) Res: (G(B))

Res: (B)

i·I BI(!

IRes~

0;.)

(5.12.2)

-H d T (BI(!)

Res: (lnd: (BI(!))

6. Basic Rickard Equivalences between Brauer Blocks 6.1. This section is devoted to a special type of Rickard equivalences which generalizes the basic Morita equivalences between blocks (cf. 3.7). To our knowledge, Michel Broll(~ is the first who guessed the existence of some special kind of Rickard equivalences which should appear underlying the so-called Deligne-Lustzig induction, for finite Chevalley groups in characteristic different from p, in some well chosen situations ([3], § 6, [4], § 6C and [5], § 4); according to him, the p-adic cohomology of Deligne-Lusztig varieties should

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121

provide us with D(G x G')-modules which restricted to any p-subgroup of G x G' would have a stable C)-basis; moreover, the corresponding pairs of blocks foreseen by Broll(~ have isomorphic defect groups and equivalent Brauer categories ([2] § 1). Jeremy Rickard proves in [17] that indeed the proposed p-adic cohomology furnishes the expected kind of D(G x G')-modules and in [18] that, assuming the equivalence of the Brauer categories for the principal blocks, this special kind of Rickard equivalences - which he denominates differently - is inherited locally (Le. between the corresponding blocks in the centralizers of the p-subgroups). 6.2. As a matter of fact, the following weaker hypothesis suffices to get this special situation between Rickard equivalent blocks. We keep all the notation of Section 5 and assume that M" induces a Rickard equivalence between band b'. We say that this equivalence is basic if, considering the action of P" x P" on 5" by left and right multiplication and denoting by Do(P") the diagonal subgroup of P" x P", both (Ker(a) x Ker(a)) . Do(P") and (Ker(a') x Ker(a')) . Do(P") stabilize C)-bases of 5"; obviously, a sufficient condition is that 5" had a (P" x P")-stable C)-basis; but notice that a basic Morita equivalence is a basic Rickard equivalence and, in that case,S" needs not to have a (P" x P")-stable C)-basis. Since we assume that the restriction of M" to c)(G x 1) and to C)(l x G') are both projective, the restriction of N" to Ker(a) = P"

n (l x

G') and Ker(a') = p"

n (G x 1)

(6.2.1)

are both projective too and therefore Ker(a) x Ker(a) and Ker(a') x Ker(a') act freely in their corresponding stable C)-basis; in particular it follows from Weiss' Theorem on permutation modules ([23], Theorem 2; see also [19], Theorem 2) that if C) has characteristic zero, the Rickard equivalence induced by M" is basic if and only if P and P' stabilize respectively C)-bases of Ind a (5") and Indal (5") by conjugation. 6.3. From now on, we assume that M" induces a basic Rickard equivalence between band b'. Unfortunately, here we cannot relate directly the local categories as we do in Theorem 3.6, but only the so-called Brauer categories of band b' (cf. [2], § 1). Recall that one goes from the local to the Brauer categories of b by replacing the local pointed groups Qo on A = c)Gb by the b-Brauer pairs (Q, b(8))), where b(8) is the block of CG(Q) such that we have b(8)BrQ(8) = BrQ(8) in (c)G)(Q) ~ kCG(Q) ([10], 2.9.2), keeping between them all the group exomorphisms in the local category; that is to say, if (Q, f) and (R, g) are two b-Brauer pairs then a group exomorphism from

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R to Q belongs to the Brauer category if and only if it is induced by x E G such that (RE)X C Q 0 for suitable local points E of Rand e5 of Q on A such that beE) = g and b(e5) = f.

6.4. Let Q 0 be a local pointed group on A, (Q 0, Q~I/' Q~, ) a local tracing triple on A, A" and A' and R a normal subgroup of Q, and denote respectively by r : Q" ~ Q and r' : Q" ~ Q' the restrictions of a and a'. It is not difficult to see that our basic hypothesis implies that Q stabilizes an (9-basis of Ag by conjugation and, consequently, that equality 4.8.3 applies; hence the unity element is primitive in Co(Ag(R))Q or, equivalently, Br~ (8) is contained in a point of Q on A(R), which actually is a local point. On the other hand, always from our basic hypothesis, statement 2.8.1 suitably modified replacing (9 by D applies to the interior D Q" -algebra A;I/I8l~ Res,' (A~,) and to the surjective group homomorphism r : Q" ~ Q. It follows quite easily that: 6.4.1. There is a section fl : R ~ r-1(R) of the restriction of r such that the interior DCQ(R)-algebra exoembedding - 01/ 0'

~

h g ' (R): Ag(R)

~

"

(Ind, (Aol/

18l~

I

Res,,(Ao,)))(R)

factorizes via e"R(A;1/ 0~ Resr,(A~,)) in an interior DCQ(R)-algebra exoembedding J-L : Ag(R) ~ " I ' h- g ~ Ind,!, (Aol/(RJ-L) 0k Res,~ (Ao,(RJ-L )))

=

where fl' r ' 0 fl, RJ-L and RJ-L' are respectively the images of fl and fl', and rJ-L ; CQI/(RJ-L) ~ CQ(R) and r~ ; CQI/(RJ-L) ~ CQ,(RJ-L')are respectively the restrictions of rand r'. Notice that fl' is injective since {O} #- A;I/(RJ-L) ~ (A;I/(RJ-L n Ker(r ' ))) (RJ-L) implies RJ-L n Ker(r ') = (1, 1). 6.5. Now, as in 5.7, choose P;I/ in such a way that (Py , P;I/' P;,) is a local tracing triple on A, A" and A'; applying statement 6.4.1 to this triple, with R = P, we get once for ever a section A : P" ~ P of a and then it can be proved with the notation of 6.4 that: 6.5.1 For any x E G such that (Qo)X C Py there is x' E G' such that (Q")(x,x') C P", (Q', b(e5')Y' c (P', bey')) and A(U X) = fl(u)(x,x') for any u ER,

but we know nothing about (e5 ' )(X,x') and (e5')x'. In particular, for any local pointed group Qo on A such that Qo C Py , we can choose a local tracing

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triple(QD,Q~JI,Q~,)on

A,A" and A' such that Q" C pll, (QI,b(O'» C and the restriction of ).. to Q fulfills condition 6.4.1 for R = Q; then applying again statement 6.5.1 to those choices, it is not difficult to prove the following result. We set )..' = a ' 0).. and, for any b-Brauer pair (Q, f) such that (Q, f) c (P, b(y», we denote by QA' the image of Q by )..' in / pi and by fA ' the block of Cc,(QA ) such that (Pi, b(y'»

(QA', fA') C (pi, b(y'».

(6.5.2)

Theorem 6.6. With the hypothesis and the notation above, the group homomorphism )..' : P ----+ pi is bijective and induces an equivalence between the Brauer categories of band b' mapping any b-Brauer pair (Q, f) contained in (P, b(y» on (QA', fA').

6.7.

Finally, we claim that in Theorem 6.6, the blocks of ] of CC(Q) and ]A' of Cc,(QA') over k are basically Rickard equivalent too. Indeed, for any b-Brauer pair, we can choose a G-conjugate (Q, f) such that (Q, f) c (P, b(y» and that Cp(Q) is a defect group of f; then it is not difficult to prove from Theorem 6.6 that C p' (QA ' ) is a defect group of fA'; in particular, if E and EA' are respectively the local points of R = Q . C p (Q) on A and of RA' = QA ' . Cpl(QA') on A' such that RE C Py and R;~, C p~, then

A E (Q) and A' A' (QA ' ) are respectively source algebras of ] and ]A'. On the E other hand, choosing a local tracing triple (RE' R~JI' R~, ) on A, A" and A' and applying statement 6.4.1 to the normal subgroup Q of R, statement 6.5.1 allow us to modify eventually our choices in such a way that we have )"(R) C R" C p" and (RA', b(E A' » C (R', b(E /» C (pi, b(y'» (6.7.1) and an interior DC p (Q) -algebra exoembedding AE(Q) ----+ Indr(A~/I(QA) 0k Resrl(A~I(QA'»)

(6.7.2)

where QA and QA' are respectively the images of Q by ).. and )..1, and r : CR',(QA) ----+ Cp(Q) and r ' : CR',(QA) ----+ CR'(QA') the restrictions of a and a'. 6.8. Actually, since the blocks b( EA') and b( E') are nilpotent ([2], § 1), the inclusion of b-Brauer pairs in 6.7.1 imply I (R A' \A' eRE' c Py'I and CRA,(Q A' ) = CR,(Q A' ) = Cp'(Q A' )(6.8.1) Moreover, although R~/1 needs not be contained in P;/I' there is (x, x') in G x G' such that (R~/I)(X,x') C P;/1 and therefore, it follows from our basic

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hypothesis that A~I/ (QA) is a simple k -algebra, so that A~I/(QA) ~ Endk(M;~,)

(6.8.2)

where M;~, is a simple A~I/(QA)-module, in particular, M;~, becomes a DC R" (QA) - module and although it needs not be indecomposable of vertex CRI/(QA') it can be proved, from exoembedding 6.7.2 and Theorem 5.5, the following result

Theorem 6.9. With the hypothesis and the notation above, there is a subgroup T" of CRI/ (QA) and an indecomposable direct summand N'Q of

Res~~I/(QA)(M;~,)

of vertex T" such that r(T") = Cp(Q),

r'(T")

C p' (QA') and we have an interior DC p (Q) -algebra exoembedding. ~

I

-11

A'

AE(Q) ~ Indp(SQ 0k Resp,(AEA,(Q »)

(6.9.1)

where SQ = Endk(N'Q) and p : T" ~ Cp(Q) and pi: T" ~ Cpl(QA ' ) are respectively the restrictions of rand r ' . In particular, a suitable indecomA'

posable direct summand of Ind~~(Q)XCcl(Q ) (N'Q) induces a basic Rickard equivalence between the blocks j of kCc(Q) and jA ' of kCCI(QA').

Remark 6.10. If N'Q lifts to an <9-free endopermutation <9T" -module N'Q (i.e. such that T" stabilizes an <9-basis of End() (N'Q) too) then Rickard's construction in [18], §5 can be easily generalized to prove that an indecomposable direct summand of A'

I dCc(Q)xCc,(Q )(N" ) n TI/ QA

induces a basic Rickard equivalence between f and fA'.

Remark 6.11. We know that RA' = R' whenever R maps surjectively onto Op(Nc(R E )/ R . Cc(R» ;

otherwise we know nothing about the inclusion R A' C R'.

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References [1]

Laurence Barker, G -algebras, Clifford Theory and the Green Correspondence, 1. Algebra 172 (1995), 335-353.

[2]

Michel Broue, Les £-blocs des groupes GL(n, q) et U(n, q2) et leurs structures locales, Asterisque 133/134 (1986), 159-188.

[3]

Michel Broue, Isometries parfaites, types de blocs, categories derivees, Asterisque 181/182 (1990), 61-92. Michel Broue, Equivalences of blocks of group algebras, Finite dimensional algebras and related topics (V. Dlab et aI., eds.), Proc. Internal. Conference Representations of AIgebras and Related Topics, Ottawa, Canada, 1992, Kluwer, 1994, 1-26. Michel Broue, Rickard Equivalences and Block Theory, Groups '93 GalwaySaint Andrews, London Math. Soc. Lecture Notes 211, Cambridge Univ. Press, 1995 , 58-79. Markus Linckelmann, Derived equivalence for cyclic blocks over a p-adic ring, Math. Z. 207 (1991), 293-304.

[4]

[5]

[6] [7]

Markus Linkelmann, The isomorphism problem for blocks with cyclic defect groups, Invent. Math. 125 (1996), 265-283.

[8]

LIuis Puig, Local Fusions in Block Source AIgebras, J. Algebra 104 (1986), 358-369.

[9]

LIuis Puig, Nilpotent blocks and their source algebras, Invent. Math. 93 (1988), 77-116.

[10]

LIuis Puig, Pointed Groups and Construction of Modules, 1. Algebra 116 (1988), 7-129.

[11]

LIuis Puig, Block source algebras in p-solvable groups, Letter to Morton Harris, 1993.

[12]

LIuis Puig, On Joanna Scopes' Criterion of Equivalences for Blocks of Symmetric Groups, Algebra Colloq.l (1994),25-55. LIuis Puig, On Thevenaz parameterization of interior G-algebras, Math. Z. 215 (1994),325-355. Jeremy Rickard, Morita Theory for derived categories, 1. London Math. Soc. 39 (1991), 37-48.

[13] [14] [15] [16] [17] [18]

Jeremy Rickard, Derived categories and stable equivalences, 1. Pure AppI. Algebra 61 (1989),307-317. Jeremy Rickard, Derived equivalences as derived functors, J. London Math. Soc. 43 (1991), 37-48. Jeremy Rickard, Finite group actions and etale cohomology, Pub!. Math. IHES 80 (1994),81-94. Jeremy Rickard, Splendid equivalences: derived categories and permutation modules, preprint 1994.

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[19]

Klaus Roggenkamp, Subgroup rigidity of p -adic group rings (Weiss arguments revisited), 1. London Math. Soc. 46 (1992), 432-448.

[20]

Klaus Roggenkamp and Leonard Scott, Verbal communication, Luminy, 1988.

[21]

Raphael Rouquier, From stable equivalences to Rickard equivalences for blocks with cyclic defect, Groups '93 Galway-Saint Andrews, London Math. Soc. Lecture Notes 211, Cambridge Univ. Press, 1995.

[22]

Leonard Scott, Unpublished notes.

[23]

Alfred Weiss, Rigidity of p-adic p-torsion, Ann. of Math. 127 (1988), 317332.

CNRS, Institut de Mathematiques de Jussieu 6 Avenue Sizet 94340 Joinville Le Pont France

Some Open Conjectures on Representation Theory G. R. Robinson

Introduction Several of Brauer's questions on block-theoretic invariants (see, for example, [3]) have in recent years been placed in a more conceptual setting. Some still appear unassailable, and some have been modified or extended by later authors. We discuss here some recent developments on some of these questions.

Brauer's Problem 21 Is there a function j : N --+ N such that whenever B is a block with defect group D, and B contains k ordinary irreducible characters, then we have IDI ~ j(k)? See [3] or [6]. This is a precise way of expressing the fact that a block with a large defect group should contain a large number of ordinary irreducible characters (if, as expected, the answer is positive). Little progress has been made on the most general case of this problem, though B. Kulshamrner recently proved that the answer is positive if we restrict attention to blocks of p-solvable groups. Very recently, B. Ktilshammer and I have used E. Zelmanov's solution of the restricted Burnside Problem to prove:

Theorem ([13]). If the p-block B has defect group D and satisfies the equality predicted by the Alperin-Mckay conjecture ([1]), then in terms of k(B).

IDI is bounded

In other words, the Alperin-McKay conjecture imples that Brauer's problem 21 has a positive answer.

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G. R. Robinson

The" k (B) at Most pd " Conjecture This is Brauer's problem to prove that the number of ordinary irreducible characters in a block B with defect group D is at most ID I. It seems fair to say that no general progress has been made on achieving this precise bound. Brauer and Feit proved that k(B) :::: ~IDe + 1 ([4]), and in general the quadratic nature of the bound has not been improved. Even for blocks of p-solvable groups, the question appears to be very difficult. In that case, the if a pi -group G acts faithfully on problem reduces to the k(GV)-problem: a GF(p)G-module V, does the semi-direct product GV have at most IVI conjugacy classes? Even the p- solvable case of the problem has interesting consequences for blocks of arbitrary groups. For example, Ktilshammer showed in [12] that a positive answer to the Alperin-McKay conjecture, together with a positive answer to this question for blocks of p-solvable groups implies a positive answer to the following conjecture of Olsson: Conjecture (J. B. Olsson [16]). Let B be a block with defect group D. Then ko(B) ::::[D : D '].

R. Knorr ([8], [9]) did fundamental work on the k(GV) problem. Building on that, and influenced by recent work of R. Gow [7], 1. G. Thompson and I proved: Theorem ([20]). Let the finite pi -group G act faithfully and irreducibly on the elementary Abelian p-group V. If there is some v in V such that Resgc(V)(V) has a faithful self-dual submodule (as GF(p)Cc(v)-module), then k(GV) :::: IVI. In my talk at the conference, I expressed the belief that such a v would always exist, thus confirming that the k(GV)-problem would always have a positive answer. Since the conference, there have been further developments. 1. G. Thompson furnished in [21] an example of a pair (G, V) for which there is no such vector v (the prime p is 7). On the positive side, in [20] it is proved that if p is outside a certain finite set of primes, then such a vector v always does exist. This result confirms the p-solvable case of the problem of this section for all but finitely many choices of the prime p. For this work and some extensions, see [19] and [20] Recently, B. Ktilshammer and I have been considering the problem of this section for principal blocks (presumably the most interesting case). We

Some Open Conjectures on RepresentationTheory

129

noted that an easy variant of Nagao's argument in [15] proves that whenever N is a normal subgroup of a finite group G, we have k(B~P)(G» ~ k(B{~P\N»k(GjN), where B{~P) denotes the principal p-block. In [11], Kovacs and I proved that there is a constant c such that when every B is a p-block of defect d of a p-solvable group, then k(B) ~ c d- 1pd. In trying to prove an anologous result for principal blocks of arbitrary groups, Ktilshammer and I have proved that if suffices to consider two configurations for a purported minimal counterexample G: (i)

Op'(G) = 1, E = E(G) i= 1, GjE is a solvable pi_group. For P E Sylp(G) we have that Gj ECG(P) is Abelian. Each component of

E is normal in G. (ii)

OP(F*(G» = 1, and G has a unique minimal normal subgroup V such that,letting X be the full pre-image in G of Op,(GjV), GjX has the structure of case i). Either V ~ (G),or V = F(G). If V ~ (G), then V = X. If V 1:.(G),then V = F*(G) is elementary Abelian.

This work is at a preliminary stage, and we hope to make further progress.

Conjectures of Alperin-Dade Type In [19], we conjecture the following, which is related to a p-Iocal computation of the number of weights (in the sense of Dade [5]) with respect to a normal subgroup. Conjecture. Whenever G is a finite group, N
~

N n Op(Z(G»,

A

[the number of N -projective irreducible characters in B which have defectd, lie over A, but are not Op (N ) -proj ective]= LaE~(N)/G( _l)lal+l [the number of Va -projective irreducible characters of G a which lie over irreducible characters of defect d of Va, lying over A, but not induced from Op(N)]. This generalizes the weight conjecture of Dade [5], which may already be viewed as a far-reaching generalization of the Alperin-McKay conjecture (and Alperin's weight conjecture, [2]) in part by virtue of [10]. Here, 'R(N) denotes the collection of radical p-chains of N. The numbers appearing in the right side are easily computed p-Iocally, using the results of Ktilshammer-Robinson [13] (recall that Va denotes the initial subgroup of the chain a).

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G. R. Robinson

The case N = G of this conjecture, in this formulation, would have the following consequence, which may be viewed as a complement to Brauer's height 0 conjecture: Conjecture. Let B be a block with defect group D. Then the defect of an irreducible character in B is always the defect of an irreducible character I-t of a radical subgroup U of D with CD(U) ~ U. In particular, if D is non-Abelian, then every irreducible character in B has height less than logp ([D : ZeD)]). Remark. With a little more care it is possible to show that if CJ = Va < VI < ... < Vn is such that B a contains a Va-projective irreducible character lying over the irreducible character I-t of defect d of Va, then we have d ~ ~ [logp (IVn I) + logp (IZ(Vn)l)], and CD(VO) ~ Va.

References [1]

J. L. Alperin, The main problem of block theory, Proc. Conf. Finite Groups, Academic Press, New York, 1976.

[2]

1. L. Alperin, Weight for finite groups, in: Proc. Symp. Pure Math 47, Amer. Math. Soc., Providence, 1987,369-379.

[3]

R. Brauer, Representations of Finite Groups, in Lectures in Mathematics, Vol. 1, Wiley, New York (1963),133-175.

[4]

R. Brauer, W. Feit, On the number of irreducible characters of finite groups in a given block, Proc. Math. Acad. Sci. USA 45 (1959), 361-365.

[5]

E.c. Dade, Counting Characters in Blocks, II, J. Reine Angew. Math. 448 (1994), 97-190.

[6]

W. Feit, The representation theory of finite groups, North Holland, Amsterdam, 1982.

[7]

R. Gow, On the number of characters in a block and the k(GV)-problem self-dual V, J. London Math. Soc. (2) 48 (1993), 441-451.

[8]

R. Knorr, On the number of characters in a p-block of a p-solvable group, Illinois J. Math. 28 (1984),181-210.

[9]

R. Knorr, A remark on Brauer's k(B)-conjecture, J. Algebra 131 (1990),444-450.

for

[10] R. Knorr, G. R. Robinson, Some remarks on a conjecture of Alperin, J. London Math. Soc. (2) 39, (1989),48-60. [11] L. G. Kovacs and G. R. Robinson, On the number of conjugacy classes of a finite group, J. Algebra 160 (1993), 441-460.

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[12] B. Kiilshammer, A remark on conjectures in modular representation theory, Arch. Math. 49 (1987), 366-399.

[13] B. Kulsharnmer, G. R. Robinson, Characters of Relatively Projective Modules II, J. London Math. Soc. (2) 36 (1987), 59-67. [14] B. Kiilshammer and G. R. Robinson, Alperin McKay implies Brauer's Problem 21, to appear in 1. Algebra. [15] H. Nagao, On a conjecture of Brauer for p-solvable groups, 1. Math. Osaka City University 13 (1962), 35-38. [16] J. B. Olsson, Block invariants in An, An and Sn, in: Proc. Symp. Pure Math. 47, Amer. Math. Soc., Providence, 1987,471-474. [17] G. R. Robinson, Some remarks on the k(GV)-problem, 1. Algebra 172 (1995), 159-166. [18] G. R. Robinson, Local structure, vertices, and Alperin's Conjecture, Proc. London Math. Soc. (3) 72 (1996), 312-330. [19] G. R. Robinson, Furtherreductions for the k( G V) -problem, J. Algebra, to appear. [20]

G. R. Robinson and J. G. Thompson, On Brauer's k(B)-problem, J. Algebra 184 (1996), 1143-1160.

[21] J. G. Thompson, private communication.

Department of Mathematics University of Leicester Leicester LE1 7RH England Email: [email protected]

Are All Groups Finite? Leonard L. Scott"

This paper is dedicated to Walter Feit on the occasion of his 65th birthday. Its contents were presented in part at the 1995 Ohio State finite group representation conference organized in celebration of that birthday. Primarily, the paper is a discussion of some classical and recent developments in the modular representation theory of finite groups of Lie type, and the problems which drive that theory. But there is also a philosophical thread ... An old question which arose again at the conference is the following: Are all groups finite? That is, applications and broader issues aside, if we think only of our interest in finite group theory itself, is it possible to safely ignore other groups? My viewpoint is that the answer to this question has two parts: First, in representation theory, at least, we cannot ignore the infinite complex Lie groups and their characteristic p analogs, the algebraic groups over IFp' The second part of my answer is that we can, nevertheless, hope to find understandings within finite group theory and finite dimensional algebra of ideas naturally suggested by these continuous contexts, and take them further. Let me begin by convincing you of the first part of my answer: Suppose one is considering a finite group G (IFq) of Lie type, such as the special linear group S L (n, q) of degree n with coefficients in the field IFq of q elements, q a power of a prime p. The Classification of finite simple groups asserts that almost all of the latter are variations on the finite groups of Lie type together with the alternating groups. Much earlier (1963), Steinberg [34], [35] proved all irreducible representations of G (IFq) with coeffic~ents in a finite field of characteristic p, and, thus, in the algebraic closure IFs- come by restriction from the irreducible representations over lFq of the algebraic group G(lFq ) . The latter group is, of course, quite infinite. It is the analog via the Zariski topology, of the complex analytic Lie group G(e). Moreover, the representations we need are continuous, and even "analytic", in the sense that they are locally defined by polynomial functions. Now, the theory of finite-dimensional

*

The author thanks NSF for its support.

Offprint from: Representation Theory of Finite Groups, Ed.: R. Solomon © by Walter de Gruyter & Co., Berlin' New York 1997

134

Leonard L. Scott

irreducible continuous representations of G(C) has an elegant and powerful formulation, first, in that the irreducible representations are parametrized very completely by the "theory of the highest weight" of Cartan, and, second, that the characters of these representations are known, given by the famous "Weyl character formula". See, for instance, [20] for these theories for complex semisimple Lie algebras. If we had such a parameterization and such a character formula for the finite groups G(IFq ) , not only would we would know their irreducible characters in the describing characteristic p of G (IFq), but we might learn something about the nondescribing case as well, where many analogies with the describing case have been discovered by Dipper and James [15], [16], [14], [21]. Working with the general linear group GL(n, q) they have found families of finitedimensional algebras, the q -Schur algebras, parameterized by a variable q, that control the nondescribing characteristic representation theory of the group GL(n, q) when q is taken to be a prime power (which may also be viewed as a kind of root of unity when the underlying field has positive characteristic), and which control the modular theory in the describing characteristic p when q = 1. If we knew both describing and nondescribing modular theory for the simple or nearly simple groups of Lie type, we could also hope to learn much about the maximal subgroup structure of all other finite groups [30], [2]. Indeed, this is a main organizational theme of the upcoming 1997 Newton Institute program at Cambridge. So, to summarize, it would be highly desirable to have for finite groups G (IFq) of Lie type a parameterization and character formula, as exist for the complex Lie groups G(C). Also, thanks to the work of Steinberg mentioned above, both issues for G (IFq) reduce to the corresponding problem for the algebraic group _G(JFq). Now it is time to tell you that the parameterization problem for G (IFq) was solved even before Steinberg's work by Chevalley, imitating the Lie-theoretic case G(C) mentioned above. Before discussing the character formula issue, let's consider how far we have come in discussing my reply to the philosophical issue, "Are all groups finite?" The first suggestion in my reply is that we cannot ignore G(C) and G (JFq ), and I hope the initial history above of the parameterization for describing characteristic representations of G(IFq ) , through Steinberg and Chevalley, duplicating Cartan's "theory of the highest weight", is convincing evidence of the usefulness of looking at continuous and algebraic groups. The problems are easier for these more richly structured groups, and some have been solved. The second suggestion in my reply, that we can abstract from these continuous contexts, and perhaps go beyond them, is evidenced

Are All Groups Finite?

135

by what has happened to the parameterization theory since that time: First, Curtis and Richen [12], [13], [29] showed it was possible to carry out an analog of the parameterization process, suitably modified, in any finite group with an appropriate split BN pair. Second, Alperin [1] demonstrated with his celebrated conjecture (a main theme of our conference) that it was possible to formulate a version of the parameterization which makes sense for any finite group! Perhaps those involved in the Classification might also say that it was useful to know that_corresponding simple group classifications already existed for G(C) and G(lFq ) , and the uniform theory of groups of Lie type which emerged was (and is) useful in efficiently dealing with many properties of known groups, and in formulating many general concepts. The structure in the Lie-theoretic case was not at all ignored, but, as above, it was only a starting point (together with involutions and the Odd Order Paper!) for a more general (and more elaborate) theory. Let's now go to the issue of a character formula for the describing characteristic representations of G (IFq). We are, I believe, far from a result as complete as the Classification, for irreducible modular representations of G(lFq ) . The few results we have put us at t.!lebeginning of the cycle, where it is still essential to learn from G(C) and G(lFq ) . Nevertheless, it has been part of the point of view of myself and my colleagues, especially in CPS (Ed Cline, Brian Parshall, and myself), to develop a theory as purely algebraic as possible, to both try and attain a more general theory and to allow for elaborations diverging from the cleanest cases. This point of view is also important in our approach to proving the current main conjecture, due to Lusztig. Before describing it, let me describe one of CPS's main algebraic abstractions, which will at least make the Lusztig conjecture easier to explain. The discussion is largely borrowed from my exposition [32].

1.1 Highest weight categories, and examples. Fix a field k, and let e be an abelian k-category (that is, e is abelian, all Hom sets are k-modules, and multiplications of morphisms is k-linear). In all cases we will consider here, e will simply be equivalent to the category of finite dimensional modules over a finite dimensional algebra, but it may not start out looking like that. We suppose the nonisomorphic irreducible objects L(A) to be indexed by the elements A of a poset A, called weights. For simplicity we will assume A is finite here; for a more general notion (requiring only that the intervals of A be finite), the reader is referred to [7]. We will also assume for simplicity that

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all objects of e have finite length, that Hom sets between objects are finitedimensional over k, and, moreover, that Endc L(A) ~ k for each A E A. We assume that e has enough projectives, and let peA) denote the projective cover of L(A). We say that e is a highest weight category if there are objects V(A), A E A, such that (1)

has head L(A), and all other composition factors of V(A) have smaller weight than A. V(A)

(2) There is an epimorphism peA) -+ V(A) with kernel filtered by objects V (IL) with IL greater than A. These conditions imply that V (A) is the largest epimorphic image of peA) with A maximal among the weights of its composition factors. Such objects arise naturally in Lie-theoretic contexts. We call V (A) a Weylobject, since it is a Weyl module in our favorite context of characteristic p algebraic group representations. Other good names are Verma object, or simply standard object. Typically, these objects are well understood, and the main object of research is to write the irreducible objects in terms of them in the Grothendieck group (that is, to obtain their "Weyl character formula"). This is precisely what the Lusztig conjecture purports to do, with certain restrictions.

Three examples in Lie theory. (1) The example which first motivated CPS is the following: Let G = G(k) be

a semisimple, simply connected algebraic group over an algebraically closed field k of positive characteristic p (e.g. k = IFq). This is the most relevant example for finite group theory. In discussing it, I will assume some basic terminology from algebraic group theory, but the reader familiar with the basic theory of root systems and Lie algebras as found in [20] should be abl~ to follow much of it. Just keep in mind the basic example G = SL(n, IFq ) . There, T below is the group of invertible diagonal matrices over IFq, B is the group of upper triangular matrices and W is the group of n x n permutation matrices. The Lie algebra of G as a vector space is n x n matrices of trace 0, and its root spaces are just the 1- dimensional spaces with arbitrary entries in the i, j position, for fixed i f. j, and 0 's elsewhere. These are common eigenspaces for the action of T by_conjugation, and the associated homomorphism (character) mapping T to IF; is called a root in the world of algebraic groups, while an arbitrary algebraic group homomorphism from T to JF q is called a weight. Let T be a fixed maximal torus, and denote the root system of T acting on the Lie algebra of G by <1>. We choose a

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set <1>+of positive roots, and let B denote the corresponding Borel subgroup corresponding to the associated set <1>-of negative roots. The set X (T) of characters (weights) on T is partially ordered by the rule: A ~ fJ, {} nO'Ci for non-negative integers n«. We also have an induced fJ, - A = LO'E+ poset structure on the set X (T)+ of dominant weights (relative to <1>+).Fix any finite set Ao of dominant weights, let A be the (finite) set of dominant weights A for which A ~ Ao for some Ao E Ao. Then the category C of finite-dimensional G- modules (in the sense of algebraic groups) which have composition factors each with maximal T - weight in A is a highest weight category with weight poset A. The Weyl modules V(A) are obtained as linear duals of modules induced to G, in the sense of algebraic groups, from dominant weights in X (T) + extended to B. (These induced modules are all finitedimensional!) They may also be obtained by a reduction modulo p process from an irreducible module in characteristic O. As such, their decomposition into weights for T is directly obtainable from the Weyl character formula. Projecting e onto any block of G-modules also gives a highest weight category. If p ~ h, the Coxeter number of the root system, it is well-known [22] that the character formulas for all irreducible modules are deducible from those in the principal block. The weights for the latter are the dominant weights in the orbit Wp.O of 0 under the 'dot' action of the affine Weyl group Wp (defined by w. u. = w(/.l+p)-p, where p is the sum of all the fundamental dominant weights, for W E Wp and fJ, E X (T).) Also, Steinberg's tensor product theorem allows us to restrict attention to restricted weights, those with coefficients less than p when expressed in terms of certain 'fundamental' weights. Let us redefine A as the set of dominant weights which are in the orbit Wp.O and bounded above by a restricted weight in that orbit. Lusztig's conjecture may then be written chL(w.O)

=

L

(-I)f(W)-f(Y)PywO,wwo(l)ch

V(y.O),

y.OEA

for any weight w.O in A. Here y, ware in Wp and Wo denotes the long word in the ordinary Weyl group W. The terms PywO,wwo(l), are values at 1 of Kazhdan-Lusnig polynomials, which are defined in a purely combinatorial way for any pair of elements in a Coxeter group. Finally f.(w) denotes the length of w in the sense of Coxeter groups (the number of fundamental reflections in a minimal expression), and the function ch( -) just assigns an object of e to the associated element in the Grothendieck group of e. The conjectured formula would give character formulas for all irreducible G-modules, so long as p ~ 2h -3, and these would in turn give corresponding

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character formulas for any finite group G(lFq ) of Lie type associated to G, with q a power of p. (Actually, so long as p ~ h, the above formula could hold for all restricted weights, and as such would have the same implications for finite groups, for such a p. This stronger version of Lusztig's conjecture was formulated by Kato [23], who apparently originally believed it to be a consequence of the original conjecture, but the arithmetic doesn't work out that way. Let me take this opportunity to mention that the first open cases for the Lusztig conjecture occur for SL(5, 5) and SL(5,7), which I have been examining with the help of an NSF undergraduate REV student. The former case is the first possibility for the Kato and Lusztig versions to diverge.) Lusztig obtained his conjecture by analogy with his conjecture with Kazhdan [24] for complex Lie algebras, which we describe next. (2) Let 9 be a complex semisimple Lie algebra, and fix a Cartan subalgebra fJand Borel subalgebra b containing fJ. Consider the corresponding category (J of BGG. The objects are the g-modules which are h-diagonalizable with finite-dimensional weight spaces (where a 'weight' here is a l-dimensional representation for the Lie algebra h ), and with the set of nonzero weights bounded above by some finite set of weights. We will also restrict attention to the case where all weights are integral; equivalently, they belong (by identification) to the set X (T) of characters for a torus T associated to fJ; these are just the integral linear combinations of the 'fundamental' weights for fJ. It is again true that any block of such modules forms a highest weight category, and all character formulas for irreducible modules are obtainable from the principal block case. The standard objects this time are the Verma modules M A , A E X(T), obtained by tensor induction of A at the universal enveloping algebra level from b to g. We write V(A) = M A , and let L(A) denote the irreducible head of V (A). The weights A indexing irreducible modules in the principal block are just those in the orbit A = W. - 2p. (This is also the orbit of 0 under the 'dot' action, since 0 = WQ. - 2p.) They correspond bijectively to elements of the Weyl group. The Kazhdan-Lusztig conjecture (now a theorem due to Brylinski-Kashiwara [6] and Beilinson-Bernstein [5]) reads chL(w.

- 2p) = L(-l)i'(W)-i'(y)

Py,w(1)ch

V(y. - 2p),

yEW

where again Py,w(1) is the value at I of a Kazhdan-Lusztig polynomial. The similarity of this formula and the previous one is remarkable, and all the more so when one considers that the standard modules in the first case are finite-dimensional, but infinite-dimensional here. The next case, is even more

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remarkable, in that we obtain precisely the same character formula for standard objects which are not modules at all, but complexes of sheaves. (3) A key ingredient in the proof of the Kazhdan-Lusztig conjecture was the Kazhdan-Lusztig formula for the stalk dimensions of the cohomology of perverse sheaves. It can be written as a character formula in the Grothendieck group sense we are using here, and we describe it below. Let X = G/ B denote the flag variety obtained from the simply connected semisimple complex Lie group G associated to the Lie algebra 9 above, and consider the category e of perverse sheaves on X with respect to the Schubert stratification and the middle perversity [4]. (Thus a stratum is a Schubert cell S (w) = B w B / B, w E W, and a perverse sheaf is a complex of sheaves of complex vector spaces with cohomology locally constant (thus constant) and finite-dimensional on Schubert cells, with certain support conditions satisfied.) The poset is W, with its Bruhat-Chevalley order, and the Weyl objects V (w), w E W, are quite easy to describe: Yew) = isCw)!C[f(w)], the extension by o of the constant sheaf, shifted downward as a complex in the derived category by degree few) Every Weyl object Yew) has a unique irreducible quotient L(w), and the axioms for a highest weight category are satisfied [28, §5]. Though unnecessary in our discussion, it is a remarkable fact that L (w) is the downward shift by f( w) of the complex (extended by zero to X) defining Goreski-MacPherson intersection cohomology on the closure of Sew); see [25] and [33]. The Grothendieck group formula of Kazhdan-Lusztig [25] reads ch L(w. - 2p) =

L (_I)£CW)-£CY)Py,w(l) ch V(y.

- 2p),

yEW

which is identical to the form of the Verma module Kazhdan-Lusztig conjecture above. Essentially, the latter conjecture was proved through an equivalence of categories reducing it to the above formula. 1.2 Quasihereditary algebras. Every highest weight category with finite weight poset and all objects of finite length is the category of finite-dimensional modules for a quasihereditary algebra S. Indeed, CPS introduced quasihereditary algebras for this reason, and proved that, conversely, the category of modules for a quasihereditary algebra could be viewed as a highest weight category [31], [281, [7]. We will not reproduce the axioms for a quasihereditary algebra here, but note that examples include hereditary algebras and poset algebras [28], as well as all finite-dimensional algebras of global dimension

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two [17]. All quotient algebras of hereditary algebras are quasihereditary. Further Lie-theoretic examples of quasihereditary algebras include Schur algebras and q-Schur algebras, and their generalizations [7], [18]. CPS believes that understanding these algebras (and variations, with various degrees of added structure) will provide a good basis for understanding representations of algebraic groups in characteristic p, and finite groups of Lie type in describing or nondescribing characteristic. A new generalization of the quasihereditary notion, very relevant to the nondescribing characteristic case, is described in the last section of this paper. Every quasihereditary algebra S has finite global dimension. The opposite algebra Sop is quasihereditary with the same weight poset, and the S-module A(A) dual to the Weyl module VOP(A) for Sop has the following remarkable property [7; p. 98, bottom]

{k

if A = ~, otherwise. Here we have assumed, as before, that End L(A) = k to simplify the statement. Using this property, and an Euler characteristic argument of Delorme, it is possible to understand why the character formulas in each of the above three cases have such a remarkably similar appearance. Moreover, by tracking in the abstract setting a version (due to MacPherson, see [33]) of the arguments used to prove the Kazhdan-Lusztig formula in the perverse sheaf case, CPS was able to provide [8] (see also [9] and [10]) the following reductions. The 'length' f(A) of a weight A = w.O below is the number of simple reflections in a reduced expression for w. Extn(V(J,l), A(A» =

o

Theorem (The CPS reductions). In each of the three examples above, the Lusrtig conjecture or its analog (the Kazhdan-Lusztig conjecture, or Kazhdan-: Lusztig formula) is equivalent to each of the following statements: (1) For each A,

u. E A, Ext l (V (J,l), L(A» =j:.0 => f(A) - f(J,l) ==1 (mod

2).

(2) For each A E A, and each weight A' adjacent to A (in the sense that the affine Weyl group or Weyl group element associated to A' is obtained from that associated to A by right multiplication by a simple reflection), we have Ext! (L(A'), L(A») =j:.O. (By a duality principle, one may take here A' < A. or A' > A). (3) For each A, J,l E A, the natural map Ext! (L(J,l), L(A» ---+Ext! (V(J,l), L(A» is surjective.

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The most promising of these reductions is perhaps the second one, though each has its own advantages. When the Lusztig conjecture is true, versions of 1) and 3) hold with the number 1 replaced by n throughout (two replacements in 1) ), and dual versions hold using A(/L), cf. [8]; see also [9]. One may study such conditions purely from the point of view of finite dimensional quasihereditary algebras, though they are far from giving us a description up to isomorphism (or a suitable weaker invariant) of the algebras involved. Until we get that far, we are somewhat in the position of trying to prove deep properties of finite simple groups without knowing what all the simple groups are.

1.3 The Lusztig program. Recently, George Lusztig [27] has formulated an attack on his own conjecture organized around the theory of quantum groups. Work by himself and Kazhdan relates a Lusztig-type conjecture for quantum groups at a root of unity to its validity in a category of 'negative level' representations for affine Kac-Moody Lie algebras. The latter conjecture, at least for simply laced root systems, is settled by work of Kashiwara and Tanisaki (also claimed by Casian, who acknowledges his original proof was in error) by reduction to a category of perverse sheaves on a generalized flag variety. Ignoring the non simply-laced case difficulties", to complete the chain, one requires a reduction from quantum groups at a pth root of unity to algebraic groups in characteristic p. This has been provided for all types by Anderson, Jantzen and Soergel [3] for p sufficiently large, depending on the individual root system and its rank. Unfortunately, no specific bound whatsoever is known for p as of this writing. The problem is that p must stay away from divisors of the index in a maximal order of a certain algebra over ::E,and the algebra is constructed so indirectly that very little information on the index is available. As Jantzen himself reported at the 1994 Banff conference, this situation is simply not acceptable to finite group theory. While CPS thinks highly of the AJS work, we regard the Lusztig conjecture as open and continue to work on it.

1.4 Stratified algebras.

Another aspect of the situation is that CPS wants a theory sufficiently general to be appropriate for nondescribing characteristic, in the spirit of the Dipper-James work on the q-Schur algebra. Already there has been work by Dipper and others (see [19]) dealing with groups other than 1 These difficulties have apparently now been handled by Kashiwara-Tanisaki.

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the general linear group. While it may be that quasihereditary algebras are involved in these cases, at least in favorable characteristics, CPS conjectures a role for a slightly more general kind of algebra. This new generalization is called a stratified algebra [11]. I describe first the most basic types of stratified algebras, the algebras with a standard stratification, which are quite close to quasihereditary algebras, and one weaker notion, algebras (whose module category is) equipped with a stratifying system. The description is quite easy to do in both cases if we just think about how we are to relax the corresponding notion (1.1) of a highest weight category: First, we relax the condition that the weights A form a poset, requiring only that they form a quasiposet, so that two weights x and IJ, may satisfy ).."::s:IJ, and IJ, ::s:).."without being equal. The equivalence classes thus obtained do themselves naturally form a poset A, and we let X denote the element of A associated with X E A. Next, in condition 1) for a highest weight category, we require only that all composition factors L(IJ,) of the standard object V()"") satisfy IJ, ::s:)..".(So that L()"") may appear twice or more, along with other composition factors L(IJ,) with X = fl.) The second condition 2) is kept in the "standard" case, and that completes the definition for that case. In the "stratifying system" case the inequality in 2) is relaxed to allow equality, but other relaxations are made as well: We just assume we have a system of objects V ()..,,)and given projective objects P()..,,)mapping onto each of these with kernel filtered by V (IJ,)'s with IJ, ~ )..". We do not require that V ()..,,)have an irreducible head, or that the quasiposet index the irreducible modules. As a replacement for the latter, we do insist that every irreducible module appear in the head of some V ()..,,). Condition 1) is replaced by the requirement that there are no nonzero homomorphisms from P()..,,)to V(IJ,) unless IJ, ::s:)..". In either case, one can prove from these conditions that the underlying algebra A has a sequence of idempotent ideals 0 = Jo C J1 C ... C I n , with n = IAI, such than Ext~/Ji (M, N) = Ext~(M, N) for all left (or right!) AIJ-modules M, N, all integers n ~ 0, and each index i. With mention of A omitted, this is the general CPS notion of a stratified algebra, with a stratification of length n. In the "standard" case, each J, I J, -1 is left projective as an A I J, -1 -module, and this is characteristic of standardly stratified algebras. The ideal need not be right projective, however. Unlike the quasihereditary case, there are algebras which are standardly left stratified but not standardly right stratified (even though the "general" notion of a stratified algebra is leftright symmetric). Apart from that, and the relaxed ordering inside standard modules, the theory of standardly stratified algebras is very close to that of

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quasihereditary algebras and highest weight categories. Note that the standard module V (A) is uniquely determined, in the "standard" case, as the largest quotient of peA) with all composition factors L(fl-) satisfying fl- ::: A. The notion of an algebra whose (left) module category has a stratifying system, and the general notion of a stratified algebra, are both weaker, but more flexible. For instance, an algebra which has a stratification of length n > 1 in the "general" sense also has one of length n - 1, obtained by removing any of the intermediate ideals in the above chain. (There is an analog ofthis statement for algebras with a stratifying system.) Also, as mentioned, the "general" notion is left-right symmetric, though this is not obvious. CPS has taken considerable trouble [11] to be able to recognize when an endomorphism algebra (e.g. a Schur algebra, q-Schur algebra, or future generalization) has a natural structure as a stratified algebra. This includes, of course, the quasihereditary case (easy to check in the presence of a standard stratification), but the generalizations are also interesting, and may be important. The recognition conditions are quite complicated, though simplify very considerably/ in special cases. They involve a kind of generalized "Specht module" theory. Rather than reproduce the conditions here, I will simply give some examples from [11], referring the reader to that paper for additional details, and further examples: Throughout k is an algebraically closed field. (I) A stratification of length 2. Let G be any nontrivial finite group. Consider the direct sum T of the trivial module and the regular permutation module over k. Then A = EndkG(T) is stratified, with IAI = 2 (the length of the stratification). Interesting cases occur already for G the cyclic group of order 2 or the Klein four group. In the first case, A is the wellknown algebra of dimension 5 which has two simple modules a and b, a b. In the second case, A is already not with projective covers band a

a

quasihereditary!. Its projective covers have Loewy layers as indicated by a

the diagrams

a

a a

band

b a

.

2 The simplifications in the preprint "Stratifying endomorphism algebras over Heeke algebras", by Du, Parshall, and Scott, over Z[q, q-l J, might even be described as dramatic.

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Leonard L. Scott

Thus, unlike the quasihereditary case, the standard object associated to a (the quotient of the first projective cover by the submodule isomorphic to the second) has a appearing with a nontrivial multiplicity. Nevertheless, inside A, the ideal generated by the idempotent associated to the second projective cover is quite nice. It is both idempotent and (left) projective. This example and the next are both standardly stratified. All standardly stratified algebras have such ideals, and their factor algebras are also standardly stratified. (2) A stratification of length 3: the dihedral group of order 8. Let G be the dihedral group of order 8, and let T be the direct sum of the trivial module, the regular module, and the two transitive permutation modules of degree 4 associated to coset spaces of the two conjugacy classes of noncentral subgroups of order 2. We take char k = 2. This time the algebra A = EndkG T is quite difficult to visualize from the given data. The general CPS approach is to try to impose a generalized 'Specht module' filtration on T, and deduce from its properties that A is indeed stratified. Without giving full details, I will at least describe the 'Specht modules' we use. Let a and b denote generators of order two for G, and put A = a - I, B = b - 1. Thus A 2 = 0 = B 2 and ABAB = BABA. We will diagram cyclic modules by indicating where a nontrivial action of A or B occurs, starting from a generator. (No arrow associated to a given node and label indicates a zero multiplication. Note that, if a node was reached by multiplication by A, then that node must be killed by A. A parallel statement holds for B.) Thus, the four transitive permutation modules making up T have diagrams

A

.,

B

A

•

• /

-,

B

•

t

t

•

•

t

•

B

t

-,

/

•

•

A

A A B B

A

• t

•

t,

• t

•

•

B

t

A

t·

B

t

•

•

•

Here the single node. is also representative of the unique irreducible (trivial) module for the group G, and the above diagrams give refined Loewy series pictures. In the CPS set-up, each of these indecomposable components of T has a 'Specht filtration', which turns into the required filtration of projective covers by standard modules for the algebra A = EndkG (T) under the contravariant

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functor HomkG( -, T). (This functor is not exact, but a filtration of T still induces a filtration of A.) Moreover, each component above has a distinguished 'Specht' submodule, though it is possible for two such 'Specht' modules to be isomorphic. For the first and second components above, the Specht submodule is the l-dimensional trivial module, while for the third and fourth component, the Specht submodule is the unique 3-dimensional submodule. The reader will observe that, when the bottom and top trivial modules are eliminated from the second component, the remainder is the direct sum of the Specht modules associated to the third and fourth component. This puts all 'Specht filtrations' in evidence. The reader may consult [11] for some general machinery to check that these filtrations are transformed into the required standard module filtrations for the indecomposable projective components of A (or can attempt a direct verification starting from the filtrations and functor we have given). CPS believes something general is happening here for all Coxeter groups, that a similar construction always leads to a nontrivial stratified algebra. We have conjectured the following:

Conjecture. Let W be a finite Coxeter group with distinguished generating set 5, 151 > 1. For J ~ 5, let denote the permutation module for kW on the cosets {WjW}WEW, Put T = E9j~s T, and A = Endrw T. Then A is stratified with respect to a quasiposet A with IAI ~ 3.

r,

For a more detailed statement of the conjecture, see [~1]. We believe the stratification arises from a stratifying system, and that A may be assumed to have a largest and smallest element, containing only one element each as equivalence classes in A, with these elements associated to Ts (the trivial module) and the sign module submodule of T¢. The stronger form of the conjecture also has as a consequence the existence of certain known resolutions, one of which is the Coxeter complex. As the final version of this paper is being readied for press, it appears that Du, Parshall and Scott will soon prove the stronger conjecture, and a q-analog. 3 CPS also expects (with lie Du) that an enlarged version of A will have a standard stratification related to the filtration of the Tr's by dual left cell modules, in the sense of Kazhdan-Lusztig, of length equal to the number of two-sided cells. The algebra A itself exhibits such a stratification in the order 8 dihedral case above. This is a special case of a Weyl group of type B. In 3 This has now been done, in the preprint described in the previous footnote.

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this case lie Du and I are working on another natural enlargement that may be quasihereditary." As mentioned above, it also seems likely that a q-analog of the conjecture holds for Heeke algebras, a possibility which makes the conjecture and stratified algebras quite relevant to nondescribing characteristic theory. This already important area of research will become even more central once the problems posed by the Lusztig conjecture itself are solved.

References [I]

J. Alperin, Cohomology in representation theory, Proc. Symp. Pure Math. 47 (1987).3-11.

[2]

M. Aschbacher and L. Scott, Maximal subgroups of finite groups, J. Algebra 92 (1985), 44-80. H. Andersen, J. Jantzen, and W. Soergel, Representations of quantum groups at a p th root of unity and of semisimple algebraic groups in characteristic p: Independence of p, Asterisque 220 (1994). A. Beilinson, J. Bernstein, and P. Deligne, Analyse et topologie sur les espaces singulares, Asterisque 100 (1982).

[3]

[4]

c. R. Acad.

[5]

A. Beilinson and J. Bernstein, Localisation des 9-modules, Paris 292 (1981),15-18.

[6]

J. Brylinski and M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent. Math. 64 (1981),387-410.

[7]

E. Cline, B. Parshall and L. Scott, Finite dimensional algebras and highest weight categories, 1. Reine Angew. Math. 391 (1988),85-99.

[8]

E. Cline, B. Parshall and L. Scott, Abstract Kazhdan-Lusztig theories, Tohoku Math. 45 (1993), 511-534.

[9]

E. Cline, B. Parshall and L. Scott, Infinitesimal Kazhdan-Lusztig theories, Contemp. Math. 39 (1992), 43-73.

[10]

E. Cline, B. Parshall and L. Scott, Simulating perverse sheaves in modular representation theory, Proc. Symp. Pure Math. 56 (1994), 63-104.

[I1]

E. Cline, B. Parshall and L. Scott, Stratifying endomorphism algebras, Mem. Amer. Math. Soc. 591, Vol. 124,1996.

[12]

C. Curtis, Irreducible representations of finite groups of Lie type, 1. Reine Angew. Math. 219 (1965),180-199.

[13]

C. Curtis, Modular representations of finite groups with a split (B,N) pair, Seminar on algebraic groups and related finite groups (A. Borel. ed.), Lecture Notes in Math. 131, Springer-Verlag, 1970,57-95.

4 This is true. The preprint by Du and Scott is entitled "The q-Schu~ algebra".

Sci.

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[14]

R. Dipper, Polynomial representations of finite general linear groups in nondescribing characteristic, Prog. in Math. 95 (1991), 343-370.

[15]

R. Dipper and G. DJames, The q-Schur algebra, 1. London Math. Soc. 59 (1989), 23-50.

[16]

R. Dipper and G. D. James, q-tensor space and q-Weyl modules, Trans. Amer. Math. Soc. 327 (1991), 251-282.

[17]

V Dlab and C. Ringel, Auslander algebras are quasi-hereditary, 1. London Math. Soc. 39 (1989), 457-466.

[18]

1. Du and L. Scott, Lusztig conjectures, old and new, I, 1. Reine Angew. Math. 455 (1994), 141-182.

[19]

R. Geck and G. Hiss, Modular representations of finite groups of Lie type in non-describing characteristic, in: Finite reductive groups: Related structures and representations (M. Cabanes, ed.), Birkhauser, 1997, 195-249.

[20]

J. Humphreys, Introduction to Lie algebras and representation theory, Grad. Texts in Math. 9, Springer-Verlag, 1970.

[21]

G. D. James, The irreducible representations of the finite general linear groups, Proc. London Math. Soc. 60 (1990), 225-265.

[22]

1. Jantzen, Representations of algebraic groups, Academic Press, 1987.

[23]

S. Kato, On the Kazhdan-Lusztig polynomials for affine Weyl groups, Adv. in Math. 55 (1985), 103-130.

[24]

D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Heeke algebras, Invent. Math.83 (1979), 165-184.

[25]

D. Kazhdan and G. Lusztig, Schubert varieties and Poincare duality, Proc. Symp. Pure Math. 36 (1980), 185-203.

[26]

G. Lusztig, Some problems in the representation theory of finite Chevalley groups, Proc. Symp. Pure Math.37 ( 1980),313-317.

[27]

G. Lusztig, Modular representations and quantum groups, Contemp. Math.82 (1989), 59-77.

[28]

B. Parshall and L. Scott, Derived categories, quasi-hereditary algebras, and algebraic groups, Mathematical Lecture Notes Series 3, Carleton University, 1988,1-105.

[29]

F. Richen, Modular representations of split BN pairs, Trans. Amer. Math. Soc. 140 (1969), 435-460.

[30]

L. Scott, Representations in characteristic p, Proc. Symp. Pure Math. 37 (1980), 319-331.

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L. Scott, Simulating algebraic geometry with algebra, I: Derived categories and Morita theory, Proc. Symp. Pure Math. 152 (1987), 271-281.

[32]

L. Scott, Quasihereditary algebras and Kazhdan-Lusztig theory, Finite dimensional algebras and related topics (V Dlab and L. Scott, eds.), Kluwer, 1994, 293-308.

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Leonard L. Scott

[33]

T. Springer, Quelques applications de la cohomologie d'intersection, Bourbaki Sem. (1981/1982).

[34]

R. Steinberg, Representations of algebraic groups, Nagoya Math. 1. 22 (1963), 33-56.

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R. Steinberg, Endomorphisms of linear algebraic groups, Mem. Amer. Math. Soc. 80, Amer. Math. Soc., Providence, R.I., 1968.

Department of Mathematics Unversity of Virginia Charlottesville, VA 22903 U.S.A. Email: [email protected]

Locally Finite Varieties of Groups and Representations of Finite Groups Sergei A. Syskin

The purpose of this talk is to attract your attention to a new area of applications of representation theory of finite groups. All definitions and explanations can be found in [2]. Let S be a variety of groups. If G is a group, then S(G) is the smallest among normal subgroups M of G with G j M E S. It is easy to see that S(G) is the subgroup generated by all values W(gl, ...• gn) where w runs over all laws of the variety S and g) • ... gn E G. Such a subgroup is called verbal. Let F be the (absolutely) free group of countable rank. Then F jS(F) is the free group of countable rank for the variety S. We say that S has a finite base of laws if S(F) is finitely generated as a verbal subgroup of F. Every finite group generates a finitely based variety (Oates-Powell Theorem). The first examples of varieties having no finite base of laws have been constructed by A. Ol'shanskii [3]. Those varieties are solvable of exponent 8 pq where p and q are relatively prime odd integers > 1. In particular, they are locally finite. For some important locally finite varieties the problem of the existence of a finite base of laws remains open. For instance (see [1], question 4.48), the following problem is still open. Question 1. Is it true that any locally finite variety of groups with abelian Sylow subgroups is finitely based? Let us consider this problem. We will call such varieties A-varieties. First, two simple facts. Lemma 1. This problem has affirmative solution finitely based.

if each solvable A-variety is

So, we may consider only solvable A -varieties.

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Sergei A. Syskin

Lemma 2. Let G be a finite group with abelian Sylow subgroups. Then the class of solvability of G is at most the number of prime divisors of IG I.

Let S be any solvable A -variety. Assume that S has no finite base of laws. Standard arguments show that S contains an infinite sequence of groups G I , G2, .. , with the following properties. (1) Every Gi, is a finite solvable group all of whose Sylow subgroups are abelian.

(2) All Gi have a restricted exponent and hence a restricted class of solvability by Lemma 2. (3)

Gi is not isomorphic to any factor of G) for i =j:. j (factor = factor-group of some subgroup).

(4)

G i is a critical group.

By definition, a group is critical if it does not lie in the variety generated by all its proper factors. Under these circumstances (Sylow sUbgroups in S are abelian) this means that every Gi has exactly one minimal normal subgroup Mi. Clearly, Mi is a p-group for some prime number p, and we may assume that p is the same for all Gi. Obviously, Pi = C (Mi) is a Sylow p -subgroup of G i. Hence G i/ Pi is a pi -group, that is of order prime to p. Moreover, we can assume that Pi = Mi' therefore

Thus, we have an ordinary faithful irreducible representation of Gi/ M i on

Mi. We have either to construct such a sequence G i satisfying (1)-(5), or to prove that it does not exist. If we are going to prove the nonexistence, we may assume something more. For example, we may assume that

(2 ') All Gi have the same exponent and the same class of solvability. Let Rand S be solvable finite groups of the same class n. We may define their derived subgroups: R(l) = R, R(i+I) = [R(i), R(i)]. We say that R and S are of the same type if the exponent of R(i) / R(i+I) equals the exponent of S(i) / S(i+I) for all i = I, ... ,n. Clearly, we can assume that (6) All Gi are of the same type.

Locally Finite Varieties

151

Using induction on the type, we can assume that our conjecture holds for the sequence G I Mi. It is a simple matter to prove that the last sequence has an infinite subsequence which is a chain of groups. Hence we may assume that (7)

Gi + I I Mi + I has a subgroup isomorphic to G i/ Mi for all i = 1, 2, ....

Of course, one may assume something else. For example, in the definition of type one can consider the series R(i) instead of RU) where R(i)1 RU+I) is the (unique) largest abelian normal subgroup of RI RU+I). Perhaps, this alternative definition is more helpful since all ordinary representations under consideration are monomial over some extension of G F(p). Thus, we may propose the following

Question 2. Let RI .::: R2 .::: '" be an infinite chain of finite solvable groups with abelian Sylow subgroups. Assume that all Ri have the same class of solvability and the same exponent. Let p be a prime which does not divide this exponent. Let Mi be a fixed faithful irreducible G F (p) Ri -module. Is it true that one of these groups, say Rj, contains a subgroup H isomorphic to Ri, i i= j, such that, for some H -submodule V of Mj, two natural semidirect products Ri Mi and V H are isomorphic? Question 3. Let p be a prime, A p the variety of all abelian groups of exponent p, and let N be a locally finite nilpotent variety such that p does not divide the exponent of N. Is it true that any subvariety of the variety ApN is finitely

based? It is known that any nilpotent variety has a finite base of laws, see [2].

References [1]

Kourovka notebook, A collection of unsolved questions in group theory, 11th edition, Novosibirsk 1990.

[2]

H. Neumann, Varieties of groups, Springer-Verlag, 1967.

[3]

A. Yu. Ol'shanskii, On the problem of a finite basis of identities in groups, Math. USSR-Izvestija 4 (1970),381-389.

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Contains articles bymanyof theparticipants at theConference on Modular Representation Theory which tookplaceunder theauspices of theMathematical Research lnstitute of theOhioStateUniversity withadditional support fromthe National Foundation.The Science conference celebrated thecareer of Professor V/alter Feitof Yale University ontheoccasion of his65thbirthday. lt alsoroughly coincided withthe centenary of thetheoryof groupcharacters. Theparticipants included mostof the leaders in thefieldof finitegrouprepresentations andthisvolumecontains and expands on manyof thestimulating lectures fromtheconference. Thisvolume is addressed to specialists andstudents in thefieldof grouprepresentation theory, including boththegeneral theory andthespecial theories related to the groupsandthefinitegroupsof Lietype.Latest representations of thesymmetric reportsaregivenon important conjectures including theLusztig conjecture, the Alperin-McKay-Dade conjectures andthekGV)problem. lmportant newresearch avenues areilluminated, including thetheory of infinite-dimensional modules forfinite groupsandthetheoryof Rickard equivalences of modulecategories, Bothexperts andgraduate students willfindthebooka fascinating introduction to a widearrayof problems newtechniques exciting and attheresearch level.

rssN0942-0363 rsBN3-11-015806-X