Diagonal $p$-permutation functors

Abstract

Let $k$ be an algebraically closed field of positive characteristic $p$, and $\mathbb{F}$ be an algebraically closed field of characteristic 0. We consider the $\mathbb{F}$-linear category $\mathbb{F}p{p}_k^{\Delta }$ of finite groups, in which the set of morphisms from $G$ to $H$ is the $\mathbb{F}$-linear extension $\mathbb{F}{T}^{\Delta }(H,G)$ of the Grothendieck group $T^{\Delta }(H,G)$ of $p$-permutation $(kH,kG)$-bimodules with (twisted) diagonal vertices. The $\mathbb{F}$-linear functors from $\mathbb{F}p{p}_k^{\Delta }$ to $\mathbb{F}-Mod$ are called {\em diagonal $p$-permutation functors}. They form an abelian category ${\mathcal{F}}_{p{p}_k}^{\Delta }$. We study in particular the functor $\mathbb{F}{T}^{\Delta }$ sending a finite group $G$ to the Grothendieck group $\mathbb{F}T(G)$ of $p$-permutation $kG$-modules, and show that $\mathbb{F}{T}^{\Delta }$ is a semisimple object of ${\mathcal{F}}_{p{p}_k}^{\Delta }$, equal to the direct sum of specific simple functors parametrized by isomorphism classes of pairs $(P,s)$ of a finite $p$-group $P$ and a generator $s$ of a $p^{\prime }$-subgroup acting faithfully on $P$. This leads to a precise description of the evaluations of these simple functors. In particular, we show that the simple functor indexed by the trivial pair $(1,1)$ is isomorphic to the functor sending a finite group $G$ to $\mathbb{F}{K}_0(kG)$, where $K_0(kG)$ is the group of projective $kG$-modules.