On the restriction of cross characteristic representations of ^2F_4(q) to proper subgroups

TL;DR

This paper proves that nontrivial representations of Ree groups of type ^2F_4(q) in odd characteristic become reducible when restricted to any proper subgroup, and classifies certain irreducible restrictions.

Contribution

It establishes the reducibility of all nontrivial representations of ^2F_4(q) on restriction to proper subgroups and classifies specific irreducible restrictions in odd characteristic.

Findings

Nontrivial representations restrict reducibly to proper subgroups.

Complete classification of certain irreducible restrictions in odd characteristic.

Identification of all triples (K, V, H) with irreducible restrictions.

Abstract

We prove that the restriction of any nontrivial representation of the Ree groups $^2F_{4}(q), q=2^{2n+1}\geq8$ in odd characteristic to any proper subgroup is reducible. We also determine all triples $(K, V, H)$ such that $K \in \{^2F_4(2), ^2F_4(2)'\}$, $H$ is a proper subgroup of $K$, and $V$ is a representation of $K$ in odd characteristic restricting absolutely irreducibly to $H$.