On Alperin's conjecture and functorial equivalence of blocks

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 $\mathbb{F}$.