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

TL;DR

This paper develops a characteristic-free algorithm to decompose tensor products of restricted simple modules for SL_3 over fields with characteristic p ≥ 5, revealing all indecomposables are rigid in this case.

Contribution

It introduces a new algorithm for tensor product decomposition in characteristic p ≥ 5 and demonstrates rigidity of indecomposables, extending previous work for p<5.

Findings

All indecomposable summands are rigid for p ≥ 5

The algorithm is characteristic-free and applicable to tensor products of SL_3 modules

Contrasts the p ≥ 5 case with the p=3 case where indecomposables are not all rigid

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.