Rational rigidity for E_8(p)

TL;DR

This paper establishes the existence of rationally rigid triples in E8 over finite fields, demonstrating their role as Galois groups over rationals, and explores related symmetries and subgroup classifications.

Contribution

It proves the existence of rationally rigid triples in E8 in good characteristic and links these to Galois groups over the rationals, revealing new symmetries and classifications.

Findings

Existence of rationally rigid triples in E8 in good characteristic.

These triples realize E8 as Galois groups over Q.

Classification of overgroups containing regular unipotent elements.

Abstract

We prove the existence of certain rationally rigid triples E8 in good characteristic and thereby show that these groups over the prime field occur as Galois groups over the field of rational numbers. We show that these triples give rise to rigid triples in the algebraic group and prove that they generate an interesting subgroup in characteristic 0. As a byproduct we derive a remarkable symmetry between the character table of a finite reductive group and that of its dual group. We also give a classification of possible overgroups in exceptional groups containing regular unipotent elements.