Paul Fong

Weights for symmetric and general linear groups

Abstract An important feature of the theory of finite groups is the number of connections and analogies with the theory of Lie groups. The concept of a weight has long been useful in the modular representation theory of finite Lie groups in the defining characteristic of the group. The idea of a weight in the modular representation theory of an arbitrary finite group was recently introduced in Alperin (Proc. Sympos. Pure Math. 41 (1987, 369–379), where it was conjectured that the number of weights should equal the number of modular irreducible representations. Moreover, this equality should hold block by block. The conjecture has created great interest, since its truth would have important consequences—a synthesis of known results and solutions of outstanding problems. In this paper we prove the conjecture first for the modular representations of symmetric groups and second for modular representations in odd characteristic r for the finite general linear groups. In the latter case r may be assumed to be different from the defining characteristic p of the group, since the result is known when r is p. The well-known analogy between the representation theory of the symmetric and general linear groups holds here too.

Brauer trees in classical groups

where the prime Y for the modular representation theory is distinct from the prime p dividing q and p is odd. The possible graphs occurring as Brauer trees of finite classical groups were described by Feit [S]. In this paper we complete his description by identifying the vertices with characters. An explicit description of Brauer trees for CL,(q) was given in [S]. We may suppose r > 2, since the trees have trivial structure for r = 2. The Jordan decomposition of characters is compatible with blocks and induces graph isomorphisms of Brauer trees for cyclic blocks. This reduces the problem to one of constructing the projective indecomposable characters in a cyclic block B where the non-exceptional characters of B are unipotent characters in the sense of Deligne and Lusztig. The projective indecomposable characters in such unipotent cyclic blocks are most readily constructed by Frobenius induction from proper subgroups. In the context of classical groups, this effectively means Harish-Chandra induction from subparabolic subgroups. Despite this relative paucity of means, the tree of B can be determined using combinatorial arguments on the partitions or symbols labeling the unipotent characters in B. In every case the tree has the form M....-.(YJ-*...OL 02 ai T, 72 1,

A characterization of the Mathieu group

In this paper we give a characterization of the simple Mathieu group 3J112 of order 95,040. The character table of 9RM2 was computed by Frobenius L6], and from his results it can be immediately seen that there exists an element F in 9I12 of order 8 such that (i) the cyclic subgroup generated by F is self-centralizing, (ii) F is conjugate to its odd powers. Elementary arguments show that there is then a Sylow 2-subgroup

ON PERFECT ISOMETRIES AND ISOTYPIES IN ALTERNATING GROUPS

of 912 Of order 64, such that (i), (ii) hold in

Generalized Harish-Chandra theory for unipotent characters of finite classical groups

3. We are thus led to a consideration of 2-groups 3 of order 64 in which (i), (ii) hold, and groups ( containing

The blocks of finite general linear and unitary groups

3 as a Sylow 2-subgroup. Our main result is the following:

Groups with a (B, N)-Pair of Rank 2. II.

Perfect isometries and isotypies are constructed for alternating groups between blocks with abelian defect groups and the Brauer correspondents of these blocks. These perfect isometries and isotypies satisfy additional compatibility conditions which imply that an extended Broue conjecture holds for the principal block of an almost simple group with an abelian Sylow psubgroup and a generalized Fitting subgroup isomorphic to an alternating group. Let G be a finite group and let 0 be a complete discrete valuation ring with field of quotients K of characteristic 0 and residue class field k of characteristic p > 0. We suppose that K contains a primitive IGI-th root of unity. In [4, (6.1)] Michel Broue posed the following Isotypy Conjecture. Let e be a block of OG with abelian defect group D and let f be the Brauer correspondent of e in (9NG (D). Then e and f are isotypic blocks. If G has an abelian Sylow p-subgroup, then the conjecture can be posed for the principal block of 0G. In this case the authors have shown that the conjecture holds provided an extended conjecture holds for the principal block of almost simple groups with an abelian Sylow p-subgroup (see [10, (5E)]). In this paper the isotypy conjecture for an arbitrary block with abelian defect group is proved for alternating groups. In addition, the extended conjecture is proved for the principal p-block of almost simple groups with abelian Sylow p-subgroups and generalized Fitting subgroup isomorphic to an alternating group. We recall the basic definitions. Let -: 0 k be the canonical quotient mapping and let -: OG kG be the induced 0-algebra homomorphism of the group algebras. In particular, -: e -*e induces a bijection between central idempotents of OG and kG. If e is a block idempotent of OG, let KGeMod be the category of left KGe-modules of finite type and let ZK (G, e) be the Grothendieck group of KGeMod. Let Gv be the set of irreducible characters of G over K. We identify R.K (G, e) with the free abelian group on (G, e)v = {X E Gv X(ge) = X(g) for all g E G}. Let CF(G, K) be the K-space of K-valued class functions on G, and let CF(G, e, K) be the K-subspace of class functions ae in CF(G, K) such that ao(ge) = a(g). The Received by the editors March 26, 1996. 1991 Mathematics Subject Classification. Primary 20C15, 20C20; Secondary 20C30. The first author was supported in part by NSF grant DMS 9100310. The second author was supported in part by NSA grant MDA 904 92-H-3027. ?1997 American Mathematical Society

The blocks of finite classical groups.

The Harish-Chandra theory for a finite group G of Lie type states the following: If p is an irreducible character of G, then there is a parabolic subgroup P of G, a Levi decomposition Lb’ of P, and a cuspidal character II/ of the Levi subgroup such that p is a constituent of the induced character indg(

Solvable groups and modular representation theory

) of the pullback

On perfect isometries and isotypies in finite groups

of