On the Galois symmetries for the character table of an integral fusion category

TL;DR

This paper explores Galois symmetries in the character tables of integral fusion categories, generalizing Burnside's theorem and revealing zero entries in non-invertible object rows.

Contribution

It introduces a natural Galois symmetry group for integral fusion categories and extends classical representation theory results to this broader context.

Findings

Integral fusion categories with rational structure constants have Galois symmetry groups.

Rows for non-invertible objects in character tables contain zero entries.

Generalization of Burnside's theorem to fusion categories.

Abstract

In this paper we show that integral fusion categories with rational structure constants admit a natural group of symmetries given by the Galois group of their character tables. We also generalize a well known result of Burnside from representation theory of finite groups. More precisely, we show that any row corresponding to a non invertible object in the character table of a weakly integral fusion category contains a zero entry.