On Donkin's Tilting Module Conjecture III: New Generic Lower Bounds

TL;DR

This paper proves several longstanding conjectures in the representation theory of reductive algebraic groups for primes greater than or equal to twice the Coxeter number minus four, significantly expanding the cases where these conjectures hold.

Contribution

The authors establish affirmative answers to key conjectures in the field under a new uniform prime bound, extending the validity to infinitely many new cases.

Findings

Confirmed Donkin's Tilting Module Conjecture for p ≥ 2h-4.

Proved the Humphreys-Verma Question under the same prime bound.

Established that certain tensor products are tilting modules for large enough primes.

Abstract

In this paper the authors consider four questions of primary interest for the representation theory of reductive algebraic groups: (i) Donkin's Tilting Module Conjecture, (ii) the Humphreys-Verma Question, (iii) whether $\operatorname{St}_r \otimes L(\lambda)$ is a tilting module for $L(\lambda)$ an irrreducible representation of $p^{r}$-restricted highest weight, and (iv) whether $\operatorname{Ext}^{1}_{G_{1}}(L(\lambda),L(\mu))^{(-1)}$ is a tilting module where $L(\lambda)$ and $L(\mu)$ have $p$-restricted highest weight. The authors establish affirmative answers to each of these questions with a new uniform bound, namely $p\geq 2h-4$ where $h$ is the Coxeter number. Notably, this verifies these statements for infinitely many more cases. Later in the paper, questions (i)-(iv) are considered for rank two groups where there are counterexamples (for small primes) to these questions.