Factoring tilting modules for algebraic groups

TL;DR

This paper proves that tensoring the Steinberg module with a minuscule module yields an indecomposable tilting module, and explores implications for the structure of tilting modules near Steinberg weights in algebraic groups.

Contribution

It establishes a new, simple proof that the tensor product of the Steinberg and minuscule modules is indecomposable tilting, with consequences for tilting module decompositions.

Findings

Tensor product of Steinberg and minuscule modules is always indecomposable tilting.

Tilting modules near Steinberg weights decompose into tensor products of simple modules under certain conditions.

New insights into the structure of tilting modules in algebraic groups.

Abstract

Let G be a semisimple, simply-connected algebraic group over an algebraically closed field of characteristic p > 0. We observe that the tensor product of the Steinberg module with a minuscule module is always indecomposable tilting. Although quite easy to prove, this fact does not seem to have been observed before. It has the following consequence: If p >= 2h-2 and a given tilting module has highest weight p-adically close to the r-th Steinberg weight, then the tilting module is isomorphic to a tensor product of two simple modules, usually in many ways.