Weight conjectures for fusion systems

Radha Kessar,Markus Linckelmann,Justin Lynd,Jason Semeraro

doi:10.1016/j.aim.2019.106825

Abstract

Many of the conjectures of current interest in the representation theory of finite groups in characteristic p are local-to-global statements, in that they predict consequences for the representations of a finite group G given data about the representations of the p-local subgroups of G. The local structure of a block of a group algebra is encoded in the fusion system of the block together with a compatible family of Külshammer-Puig cohomology classes. Motivated by conjectures in block theory, we state and initiate investigation of a number of seemingly local conjectures for arbitrary triples (S,F,α) consisting of a saturated fusion system F on a finite p-group S and a compatible family α.

Highlights

Throughout this paper we fix a prime number p and an algebraically closed field k of characteristic p
If B is the principal block of kG, S is a Sylow p-subgroup, F = FS(G), and all the classes αQ are trivial
If B is a block of a finite group algebra kG, we denote by k(B) the number of ordinary irreducible characters of G associated with B

Summary

Introduction

Throughout this paper we fix a prime number p and an algebraically closed field k of characteristic p. Let (S, F, α) be a triple consisting of a finite p-group S, a saturated fusion system F on S, and a family α = (αQ)Q∈Fc of classes αQ ∈ H2(OutF (Q); k×), for any F -centric subgroup Q. of S, such that the family α is F-compatible in the sense of Definition 4.1 below. Theorem 1.2 shows that AWC implies an equality (for arbitrary fusion systems) of two numerical invariants dual to each other in the sense that one is obtained by summing over conjugacy classes of p-groups and the other by summing over irreducible characters. If (S, F, α) is block realizable, Corollary 1.3 follows from work of Robinson and expresses the fact that a coarse version of the Ordinary Weight Conjecture is implied by AWC (see Theorem 2.4 below). In an Appendix, we collect some foundational material from work of Robinson

Conjectures

Compatible families of second cohomology classes