The Galois Alperin weight conjecture for finite category algebras

TL;DR

This paper formulates a Galois Alperin weight conjecture for finite category algebras, reducing it to the case of finite groups, and extends it to EI-categories with a blockwise version.

Contribution

It introduces a new conjecture for finite category algebras, linking simple modules and weights, and reduces the problem to finite groups, extending to EI-categories.

Findings

Formulated a Galois Alperin weight conjecture for finite category algebras.

Reduced the conjecture to the case of finite groups.

Extended the conjecture to EI-categories with a blockwise formulation.

Abstract

Let $p$ be a prime, $k$ an algebraic closure of $\mathbb{F}_p$ and $\Gamma$ the Galois group ${\rm Gal}(k/\mathbb{F}_p)$. Let $\mathcal{C}$ be a finite category and $\mathcal{O}_{\mathcal{C}}$ the $p$-orbit category of $\mathcal{C}$ defined by Linckelmann. We formulate a version of the Galois Alperin weight conjecture (GAWC) for finite category algebras stating that there exists a $\Gamma\times {\rm Aut}(\mathcal{C})$-equivariant bijection between the set of isomorphism classes of simple $k\mathcal{C}$-modules and that of the weights of $k\mathcal{O}_{\mathcal{C}}$. We reduce the GAWC for finite categories to finite groups. For $\mathcal{C}$ an EI-category, we give a partition of weights of $k\mathcal{O}_{\mathcal{C}}$ with respect to blocks of $k\mathcal{C}$ and then formulate a blockwise Galois Alperin weight conjecture (BGAWC) for $\mathcal{C}$. Similarly, we reduce the BGAWC for finite EI-categories to finite groups.