The diagonal p $p$ -permutation functor k R k $kR_k$

Let $k$ be an algebraically closed field of positive characteristic $p$. We describe the full lattice of subfunctors of the diagonal $p$-permutation functor $k{R}_k$ 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 $k{R}_k$, which are parameterized by finite $p$-groups (up to isomorphism), and of the evaluations of these particular simple diagonal $p$-permutation functors over $k$.