On Perfect Isometries and Isotypies in Alternating Groups

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 Broué 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 O 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 |G|-th root of unity. In [4, (6.1)] Michel Broué 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 ONG(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 OG. 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 ̄ : O −→ k be the canonical quotient mapping and let ̄ : OG −→ kG be the induced O-algebra homomorphism of the group algebras. In particular, ̄ : e −→ ē 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 RK(G, e) be the Grothendieck group of KGeMod. Let G ∨ be the set of irreducible characters of G over K. We identify RK(G, e) with the free abelian group on (G, e)∨ = {χ ∈ G∨ : χ(ge) = χ(g) for all g ∈ G}. Let CF(G,K) be the K-space ofK-valued class functions on G, and let CF(G, e,K) be the K-subspace of class functions α in CF(G,K) such that α(ge) = α(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. c ©1997 American Mathematical Society