Representation Theory of Finite Groups in Wiener--Hopf Factorization

TL;DR

This paper explores how symmetry in matrix functions, defined by finite group permutations, can simplify Wiener--Hopf factorization, enabling explicit solutions and relations between partial indices.

Contribution

It introduces a method to reduce the Wiener--Hopf problem's dimension using group symmetries and derives explicit factorizations in special cases.

Findings

Reduced problem dimension via symmetry

Relations between partial indices established

Explicit factorizations obtained in special cases

Abstract

We consider the Wiener--Hopf factorization problem for a matrix function that is completely defined by its first column: the succeeding columns are obtained from the first one by means of a finite group of permutations. The symmetry of this matrix function allows us to reduce the dimension of the problem. In particular, we find some relations between its partial indices and can compute some of the indices. In special cases we can explicitly obtain the Wiener--Hopf factorization of the matrix function.