The equality of generalized matrix functions on the set of all symmetric matrices

TL;DR

This paper characterizes when generalized matrix functions on symmetric matrices are equal, showing they coincide with the determinant under certain conditions, and explores their multiplicative and commutative properties.

Contribution

It provides necessary and sufficient conditions for the equality of generalized matrix functions on symmetric matrices, linking them to the determinant and analyzing their algebraic properties.

Findings

Generalized matrix functions equal the determinant under specific conditions.

The multiplicative property $d_ ext{G}^G(AB) = d_ ext{G}^G(A)d_ ext{G}^G(B)$ holds iff the function is the determinant.

Conditions for when $d_ ext{G}^G(AB) = d_ ext{G}^G(BA)$ are established.

Abstract

A generalized matrix function $d_\chi^G : M_n(\mathbb{C}) \rightarrow \mathbb{C}$ is a function constructed by a subgroup $G$ of $S_n$ and a complex valued function $\chi$ of $G$. The main purpose of this paper is to find a necessary and sufficient condition for the equality of two generalized matrix functions on the set of all symmetric matrices, $\mathbb{S}_n(\mathbb{C})$. In order to fulfill the purpose, a symmetric matrix $S_\sigma$ is constructed and $d_\chi^G(S_\sigma)$ is evaluated for each $\sigma \in S_n$. By applying the value of $d_\chi^G(S_\sigma)$, it is shown that $d_\chi^G(AB) = d_\chi^G(A)d_\chi^G(B)$ for each $A, B \in \mathbb{S}_n(\mathbb{C})$ if and only if $d_\chi^G = \det$. Furthermore, a criterion when $d_\chi^G(AB) = d_\chi^G(BA)$ for every $A, B \in \mathbb{S}_n(\mathbb{C})$, is established.