Decomposition of Tensor Products of Modular Irreducible Representations for $SL_3$: the $p \geq 5$ case

Abstract

We study the structure of the indecomposable direct summands of tensor products of two restricted simple $SL_3(K)$-modules, where $K$ is an algebraically closed field of characteristic $p \geq 5$. We give a characteristic-free algorithm for the computation of the decomposition of such a tensor product into indecomposable modules. The $p<5$ case for $\SL_3(K)$ was studied in the authors' earlier paper. In this paper we show that for characteristics $p\geq 5$ all the indecomposable summands are rigid, in contrast to the situation in characteristic 3.