Equivariant correspondences and the inductive Alperin weight condition for type $\mathsf A$

Abstract

In this paper, we establish the inductive Alperin weight condition for the finite simple groups of Lie type A, contributing to the program to prove the Alperin weight conjecture by checking the inductive condition for all finite simple groups.

The Inductive Blockwise Alperin Weight Condition for Type and the Prime

“…Now By [20,Thm. 7.1], there exists φ i, j,k Irr(N H i (Q 0,i, j )/Q 0,i, j ) which is N H i (Q 0,i, j )-conjugate to ϕ 0,i, j,k , such that (N H i (Q 0,i, j )B i ) φ i, j,k = N H i (Q 0,i, j ) φ i, j,k (B i ) φ i, j,k and φ i, j,k extends to N H i (Q 0,i, j )(B i ) φ i, j,k .…”

“…Using these, there exists a blockwise IBr( G/G) ⋊ B-equivariant bijection between IBr( G) and Alp( G), see [19] and [34]. In addition, the weights of G were classified and the inductive condition of the non-blockwise Alperin weight conjecture for simple groups of type A was established for blocks of G in [20].…”

“…3.18] holds and for the rest conditions we transfer from B to A. By Theorem 7.1 and 8.1 of [20], the condition (iv) of [23,Thm. 3.18]…”

“…5.7], it suffices to show that the strictly quasi-isolated blocks of G F are BAW-good, since Assumption 5.3 of [23] holds for G by [13,Thm. 4.1] and the unitriangularity of decomposition matrices in [15,32]; see also [20,Thm. 8.1].…”

“…Let G = SL n (ǫq) (ǫ = ±1), where SL n (−q) denotes SU n (q). In [20], the weights of G were classified and the inductive condition of the non-blockwise Alperin weight conjecture was verified for groups of type A. By the criterion given by Brough-Späth [12], the key of the verification of the inductive BAW condition for the simple group G/ Z(G) is to establish an Aut(G)-equivariant bijection between IBr(G) and Alp(G) which preserves blocks.…”