A computational study of certain Weyl modules for type $G_2$ in characteristic 2

TL;DR

This paper uses computational methods to analyze Weyl modules for type G_2 in characteristic 2, providing counterexamples to longstanding conjectures and demonstrating the capabilities of the WeylModules GAP package.

Contribution

It offers the first computational evidence against two conjectures by Donkin and showcases the utility of the WeylModules package for algebraic group representations.

Findings

Counterexamples to Donkin's conjectures in characteristic 2

Demonstration of the WeylModules package's capabilities

Potential application to other algebraic groups and characteristics

Abstract

Using the \texttt{WeylModules} \textsf{GAP} Package, we compute structural information about certain Weyl modules for type $G_2$ in characteristic $2$. This gives counterexamples to two conjectures stated by S.~Donkin in 1990. It also illustrates capabilities of the package, which can in principle be applied to Weyl modules for any simple, simply-connected algebraic group in any characteristic, subject of course to time and space limitations of computational resources.