On Donkin's Tilting Module Conjecture IV: New Frontiers

Abstract

Let $G$ be a simple, simply connected algebraic group scheme defined over $\mathbb{F}_{p}$, and let $G_{r}$ be the $r$th Frobenius kernel. Donkin's famous Tilting Module Conjecture purports that a given indecomposable injective $G_{r}$-module can be realized as the restriction of a specific tilting module. The conjecture was first stated in 1990 and withstood proof for nearly 30 years. The authors discovered a counterexample in 2019. This paper provides some indication about the rich mathematics surrounding the Tilting Module Conjecture that also discusses an older conjecture by Humphreys and Verma. New versions of the Tilting Module Conjectures are formulated given multiple families of new counterexamples. Finally, through a new calculation, the authors present further evidence that the Tilting Module Conjecture should hold for reductive algebraic groups with an underlying root system of Type $\mathrm{A}_{n}$ for all fields of characteristic $p > 0$.