The Galois action on character tables

TL;DR

This paper explores a geometric perspective on how Galois groups act on finite group character tables, extending the concept to a broader space involving maps from G^n to algebraic closures, with G acting by conjugation and S_n permuting components.

Contribution

It introduces a geometric interpretation and generalization of the Galois action on character tables to a new space involving maps from G^n, expanding understanding of Galois symmetries.

Findings

Provides a geometric framework for Galois actions on character tables

Generalizes Galois action to the space Map_G(G^n, Q)/S_n

Suggests new connections between Galois theory and representation theory

Abstract

A geometric interpretation and generalisation for the Galois action on finite group character tables is sketched. The generalisation is a Galois action on the space Map_G(G^n,\bar{Q})/S_n for each finite G, where G acts by simultaneous conjugation on the n-tuples G^n and the symmetric group S_n permutes the components.