# Crossed $S$-matrices and Fourier matrices for Coxeter groups with automorphism

## TL;DR

This paper investigates crossed S-matrices in braided G-crossed categories, reducing their computation to submatrices, and applies this to derive Fourier matrices for specific unipotent character families in dihedral and Ree groups.

## Contribution

It introduces a method to compute crossed S-matrices via de-equivariantization and applies it to obtain Fourier matrices for unipotent characters of certain finite groups.

## Key findings

- Reduced computation of crossed S-matrices to submatrices
- Derived Fourier matrices for dihedral groups with automorphism
- Derived Fourier matrices for Ree group of type {}^2F_4

## Abstract

We study crossed $S$-matrices for braided $G$-crossed categories and reduce their computation to a submatrix of the de-equivariantization. We study the more general case of a category containing the symmetric category $\mathrm{Rep}(A,z)$ with $A$ a finite cyclic group and $z\in A$ such that $z^2=1$. We give two example of such categories, which enable us to recover the Fourier matrix associated with the big family of unipotent characters of the dihedral groups with automorphism as well as the Fourier matrix of the big family of unipotent characters of the Ree group of type ${}^2F_4$.