On Donkin's Tilting Module Conjecture I: Lowering the Prime

TL;DR

This paper proves Donkin's Tilting Module Conjecture for all rank 2 semisimple algebraic groups and SL4 over algebraically closed fields of positive characteristic, introducing new techniques involving filtrations and character formulas.

Contribution

It provides a complete solution to Donkin's conjecture in specific cases and introduces novel methods like $(p,r)$-filtrations and Lusztig's character formula applications.

Findings

Confirmed Donkin's conjecture for rank 2 groups and SL4

Developed new techniques involving filtrations and character formulas

Enhanced understanding of tilting modules in positive characteristic

Abstract

In this paper the authors provide a complete answer to Donkin's Tilting Module Conjecture for all rank $2$ semisimple algebraic groups and $\text{SL}_{4}(k)$ where $k$ is an algebraically closed field of characteristic $p>0$. In the process, new techniques are introduced involving the existence of $(p,r)$-filtrations, Lusztig's character formula, and the $G_{r}$T-radical series for baby Verma modules.