On tensoring with the Steinberg representation

TL;DR

This paper investigates the conditions under which tensoring the Steinberg module with certain simple modules results in modules with good filtrations, advancing understanding of conjectures in algebraic group representation theory.

Contribution

It verifies the good filtration property for tensor products involving the Steinberg module under various conditions on the characteristic and group rank.

Findings

Confirmed the property for p ≥ 2h-4.

Established the result for all rank two groups.

Proved the case for p ≥ 3 with fundamental weights.

Abstract

Let $G$ be a simple, simply connected algebraic group over an algebraically closed field of prime characteristic $p>0$. Recent work of Kildetoft and Nakano and of Sobaje has shown close connections between two long-standing conjectures of Donkin: one on tilting modules and the lifting of projective modules for Frobenius kernels of $G$ and another on the existence of certain filtrations of $G$-modules. A key question related to these conjectures is whether the tensor product of the $r$th Steinberg module with a simple module with $p^{r}$th restricted highest weight admits a good filtration. In this paper we verify this statement when (i) $p\geq 2h-4$ ($h$ is the Coxeter number), (ii) for all rank two groups, (iii) for $p\geq 3$ when the simple module corresponds to a fundamental weight and (iv) for a number of cases when the rank is less than or equal to five.