Radically filtered quasi-hereditary algebras and rigidity of tilting modules

TL;DR

This paper investigates conditions under which tilting modules over quasi-hereditary algebras are rigid, and applies these results to analyze the radical series of tilting modules for SL_4(K) in positive characteristic.

Contribution

It introduces criteria for the rigidity of tilting modules based on subquotients and applies these to specific modules for SL_4(K) in positive characteristic.

Findings

Tilting modules are often rigid if they lack certain subquotients.

New results on the radical series of tilting modules for SL_4(K).

Conditions for rigidity depend on extension properties of composition factors.

Abstract

Let $A$ be a quasi-hereditary algebra. We prove that in many cases, a tilting module is rigid (i.e. has identical radical and socle series) if it does not have certain subquotients whose composition factors extend more than one layer in the radical series or the socle series. We apply this theorem to give new results about the radical series of some tilting modules for $SL_4(K)$, where $K$ is a field of positive characteristic.