# Conjugacy classes and centralizers for pivotal fusion categories

## TL;DR

This paper provides criteria for centralizers in braided fusion categories, extends identities on class functions, and shows that conjugacy class sum products in modular tensor categories have rational coefficients.

## Contribution

It introduces a new criterion for M"uger centralizers, extends identities on class functions, and analyzes conjugacy class sums in modular tensor categories.

## Key findings

- Criterion for M"uger centralizer established
- Extended identities on class functions in fusion categories
- Product of conjugacy class sums has rational coefficients

## Abstract

A criterion for M\"uger centralizer of a fusion subcategory of a braided non-degenerate fusion category is given. Along the way we extend some identities on the space of class functions of a fusion category introduced by Shimizu in \cite{scalg}. We also show that in a modular tensor category the product of two conjugacy class sums is a linear combination of conjugacy class sums with rational coefficients.