On Alperin's conjecture and functorial equivalence of blocks

TL;DR

This paper explores a functorial approach to Alperin's weight conjecture in modular representation theory, establishing an equivalence with the original conjecture and proving its stability in a functor category.

Contribution

It formulates a functorial version of Alperin's weight conjecture and proves its equivalence and stability, advancing understanding of block theory in modular representations.

Findings

Functorial version of Alperin's weight conjecture is equivalent to the original.

The conjecture holds stably in the category of stable diagonal p-permutation functors.

Provides new insights into the functorial nature of block invariants.

Abstract

Let $k$ be an algebraically closed field of positive characteristic $p$ and let $\mathbb{F}$ be an algebraically closed field of characteristic 0. We consider Alperin's weight conjecture (over $k$) from the point of view of (stable) functorial equivalence of blocks over $\mathbb{F}$. We formulate a functorial version of Alperin's blockwise weight conjecture, and show that it is equivalent to the original one. We also show that this conjecture holds stably, i.e., in the category of stable diagonal $p$-permutation functors over $\mathbb{F}$.