Height zero characters and Galois automorphisms

TL;DR

This paper proves a strengthened version of Brauer's height zero conjecture considering Galois automorphisms, and derives a Galois analogue of the Itô-Michler theorem, advancing understanding of block theory in finite groups.

Contribution

It introduces a Galois-equivariant version of Brauer's height zero conjecture and establishes a Galois version of the Itô-Michler theorem, answering recent conjectures.

Findings

Proved a Galois-equivariant height zero conjecture for principal p-blocks.

Derived a Galois analogue of the Itô-Michler theorem.

Answered a recent conjecture by Malle, Moretó, Rizo, and Schaeffer Fry.

Abstract

Let $G$ be a finite group and let $p$ be a prime. In this paper, we prove a strengthened version of Brauer's height zero conjecture for the principal $p$-block of $G$ that takes the action of a certain group of Galois automorphisms into account. This answers a conjecture recently proposed by Malle, Moret\'o, Rizo and Schaeffer Fry. We then use this to obtain a structural result which can be seen as a Galois version of the It\^o-Michler theorem.