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

TL;DR

This paper classifies simple diagonal p-permutation functors over fields of characteristic 0 or p, linking them to group structures and describing their evaluations in terms of conjugacy classes and defect groups.

Contribution

It determines when the essential algebra is non-zero and parametrizes simple functors using triples involving D^Δ-pairs and modules over automorphism groups.

Findings

Classification of groups with non-zero essential algebra

Parametrization of simple functors by triples (L,u,W)

Dimension of simple functors related to conjugacy classes of p-regular elements

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 $G=L\rtimes \langle u\rangle$, where $(L,u)$ is a $D^\Delta$-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 $D^\Delta$-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 $L$.