Alperin's weight conjecture, Galois automorphisms, alternating sums, and functorial equivalences

TL;DR

This paper explores the Galois Alperin weight conjecture using functorial equivalences, reformulating it with alternating sums and Grothendieck groups, and establishing conditions under which the conjecture holds.

Contribution

It introduces a functorial reformulation of the BGAWC and proves its equivalence with existing formulations, also linking functorial equivalences to the conjecture's validity.

Findings

Functorial equivalences provide new insights into BGAWC.

Reformulation of BGAWC using alternating sums and Grothendieck groups.

Functorial equivalences imply the conjecture holds for certain blocks.

Abstract

We show that functorial equivalences can offer new insight into the blockwise Galois Alperin weight conjecture (BGAWC). Inspired by Kn\"orr and Robinson's work, we first formulate the BGAWC in terms of alternating sums indexed by chains of $p$-subgroups, and we also give a functorial reformulation in the Grothendieck group of diagonal $p$-permutation functors. We prove that these formulations are equivalent. We further show that if a functorial equivalence between a block with abelian defect group and its Brauer correspondent descends to the minimal field of the block, then the BGAWC holds for that block. Finally, we prove that Galois conjugate blocks are functorially equivalent over an algebraically closed field of characteristic zero.