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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\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 parameterization 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 $\mathsf {S}_{L,u,W}$ parameterized 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 a defect group isomorphic to L.

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Diagonal p-permutation functors in characteristic p

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Diagonal p-permutation functors in characteristic p