Diagonal $p$-permutation functors in characteristic $p$

Abstract

Let $p$ be a prime number. We consider diagonal $p$-permutation functors over a (commutative, unital) ring $\mathsf{R}$ in which all prime numbers different from $p$ are invertible. We first determine the finite groups $G$ for which the associated essential algebra ${\mathcal{E}}_{\mathsf{R}}(G)$ is non zero: These are groups of the form $G=L⋊⟨u⟩$, where $(L,u)$ is a $D^{\hat{I}}”$-pair. When $\mathsf{R}$ is an algebraically closed field $\mathbb{F}$ of characteristic 0 or $p$, this yields a parametrization of the simple diagonal $p$-permutation functors over $\mathbb{F}$ by triples $(L,u,W)$, where $(L,u)$ is a $D^{\hat{I}}”$-pair, and $W$ is a simple $\mathbb{F}\mathrm{Out}(L,u)$-module. Finally, we describe the evaluations of the simple functor ${\mathsf{S}}_{L,u,W}$ parametrized by the triple $(L,u,W)$. We show in particular that if $G$ is a finite group and $\mathbb{F}$ has characteristic $p$, the dimension of ${\mathsf{S}}_{L,1,\mathbb{F}}(G)$ is equal to the number of conjugacy classes of $p$-regular elements of $G$ with defect isomorphic to $L$.