Positive Self-Dual Hopf Algebras of Galois Characters

TL;DR

This paper introduces Galois characters derived from Galois group actions on complex characters, demonstrating their structure forms a positive self-dual Hopf algebra and classifying these for general linear groups over finite fields.

Contribution

It defines Galois characters and classes, shows they form a PSH similar to classical characters, and classifies them for general linear groups over finite fields.

Findings

Galois characters form a positive self-dual Hopf algebra

Classification of Galois characters for general linear groups over finite fields

Isomorphism between Galois character PSH and tensor products of PSHs

Abstract

By using the action of certain Galois groups on complex irreducible characters and conjugacy classes, we define the Galois characters and Galois classes. We will introduce a set of Galois characters, called Galois irreducible characters, and we show that each Galois character is a non-negative linear combination of the Galois irreducible characters. It is shown that whenever the complex characters of the groups of a tower produce a positive self-dual Hopf algebra (PSH), Galois characters of the groups of the tower also produce a PSH. Then we will classify the Galois characters and Galois classes of the general linear groups over finite fields. In the end, we will precisely indicate the isomorphism between the PSH of Galois characters and a certain tensor product of positive self-dual Hopf algebras.