Globally Irreducible Weyl Modules

TL;DR

This paper characterizes all Weyl modules that remain irreducible over every field, showing they are either minuscule or related to the adjoint representation of E8, confirming a conjecture by Gross.

Contribution

It proves a converse to known irreducibility results, classifying all Weyl modules that are irreducible over all fields, as conjectured by Gross.

Findings

Weyl modules with minuscule highest weight are irreducible over all fields.

The adjoint representation of E8 is irreducible over all fields.

Any Weyl module irreducible over all fields is either minuscule or derived from E8's adjoint representation.

Abstract

In the representation theory of split reductive algebraic groups, it is well known that every Weyl module with minuscule highest weight is irreducible over every field. Also, the adjoint representation of $E_8$ is also irreducible over every field. In this paper, we prove a converse to these statements, as conjectured by Gross: if a Weyl module is irreducible over every field, it must be either one of these, or trivially constructed from one of these.