The Galois Alperin weight conjecture for finite category algebras

Abstract

Let $p$ be a prime, $k$ an algebraic closure of ${\mathbb{F}}_p$ and $\Gamma$ the Galois group $\mathrm{Gal}(k/{\mathbb{F}}_p)$. 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 $\Gamma \times \mathrm{Aut}(\mathcal{C})$-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.