A Tensor Product Factorization For Certain Tilting Modules

TL;DR

This paper extends the tensor product factorization of certain tilting modules for semisimple algebraic groups to cases with less restrictive characteristic p, using the broader concept of (p,r)-minuscule weights.

Contribution

It generalizes previous tensor product factorization results by removing the p >= 2h-2 restriction and introduces (p,r)-minuscule weights for broader applicability.

Findings

Extended tensor product factorization to smaller p values

Introduced (p,r)-minuscule weights for tilting modules

Broadened understanding of module decompositions in positive characteristic

Abstract

Let G be a semisimple, simply connected linear algebraic group over an algebraically closed field k of characteristic p > 0. In a recent paper [4], Doty introduces the notion of r-minuscule weight and exhibits a tensor product factorization of a corresponding tilting module under the assumption p >= 2h-2, where h is the coxeter number. We remove this restriction and consider some variations involving the more general notion of (p,r)-minuscule weights.