The diagonal $p$-permutation functor $kR_k$

TL;DR

This paper characterizes the subfunctor structure and simple components of the diagonal p-permutation functor over an algebraically closed field of characteristic p, revealing its composition factors parametrized by finite p-groups.

Contribution

It provides a complete description of the subfunctor lattice and simple constituents of the diagonal p-permutation functor, extending the understanding of its algebraic structure.

Findings

Lattice of subfunctors of $kR_k$ fully described

Composition factors $S_P$ parametrized by finite p-groups identified

Evaluations of simple functors over $k$ characterized

Abstract

Let $k$ be an algebraically closed field of positive characteristic $p$. We describe the full lattice of subfunctors of the diagonal $p$-permutation functor $kR_k$ obtained by $k$-linear extension from the functor $R_k$ of linear representations over $k$. This leads to the description of the ``composition factors'' $S_P$ of $kR_k$, which are parametrized by finite $p$-groups (up to isomorphism), and of the evaluations of these particular simple diagonal $p$-permutation functors over $k$.