On $(p,r)$-Filtrations and Tilting Modules

TL;DR

This paper explores the relationship between two conjectures in representation theory, showing that a weaker form of one conjecture nearly implies the other, thus deepening understanding of tilting modules and filtrations.

Contribution

It establishes a near equivalence between Donkin's Tilting Module Conjecture and a weaker version of the Good $(p,r)$-Filtration Conjecture, advancing theoretical understanding.

Findings

Tilting Module Conjecture implies one direction of the Good $(p,r)$-Filtration Conjecture.

A weaker version of the Good $(p,r)$-Filtration Conjecture nearly implies the Tilting Module Conjecture.

The results connect two major conjectures, suggesting their potential equivalence under certain conditions.

Abstract

We study the relationship between Donkin's Tilting Module Conjecture and Donkin's Good $(p,r)$-Filtration Conjecture. Our main result was motivated by a result of Kildetoft and Nakano showing that the Tilting Module Conjecture implies one direction of the Good $(p,r)$-Filtration Conjecture. We observe that the converse nearly holds; in particular a weaker version of the Good $(p,r)$-Filtration Conjecture implies the Tilting Module Conjecture.