# Steinberg squares and tensor products of tilting modules with simple modules

## TL;DR

This paper establishes a $G$-module isomorphism linking Steinberg squares and tensor products of simple and tilting modules, providing a characterization of Donkin's Tilting Module Conjecture in algebraic group theory.

## Contribution

It offers a $G$-module theoretic characterization of Donkin's Tilting Module Conjecture through an explicit isomorphism involving Steinberg modules, simple modules, and tilting modules.

## Key findings

- Isomorphism holds if and only if Donkin's Tilting Module Conjecture is true.
- Provides a new perspective on the conjecture via module-theoretic characterization.
- Connects Steinberg squares with tensor products of simple and tilting modules.

## Abstract

Let $G$ be a simple and simply connected algebraic group over an algebraically closed field $\Bbbk$ of characteristic $p>0$. We establish an isomorphism of $G$-modules between a direct sum of modules $\text{St} \otimes \text{St}$ and a direct sum of tensor products of simple modules of restricted highest weight with tilting modules that are projective over the Frobenius kernel of $G$. This isomorphism holds precisely when Donkin's Tilting Module Conjecture does, and thus can be seen as providing a $G$-module theoretic characterization of this conjecture.