Decomposition of tensor products of modular irreducible representations for $SL_3$ (With an Appendix by C.M. Ringel)

TL;DR

This paper develops an algorithm to decompose tensor products of simple modules for SL_3 in characteristics 2 and 3, revealing new non-rigid tilting modules and their structures, with implications for Schur algebras.

Contribution

It introduces a finite family of modules to express all indecomposable summands as twisted tensor products, and uncovers the first examples of non-rigid tilting modules for algebraic groups.

Findings

Finite family of modules for tensor product decomposition

Existence of non-rigid tilting modules in characteristic 3

Examples of non-rigid projective modules for Schur algebras

Abstract

We give an algorithm for working out the indecomposable direct summands in a Krull--Schmidt decomposition of a tensor product of two simple modules for G=SL_3 in characteristics 2 and 3. It is shown that there is a finite family of modules such that every such indecomposable summand is expressible as a twisted tensor product of members of that family. Along the way we obtain the submodule structure of various Weyl and tilting modules. Some of the tilting modules that turn up in characteristic 3 are not rigid; these seem to provide the first example of non-rigid tilting modules for algebraic groups. These non-rigid tilting modules lead to examples of non-rigid projective indecomposable modules for Schur algebras, as shown in the Appendix. Higher characteristics (for SL_3) will be considered in a later paper.