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

Abstract

A generalized matrix function $d_{\chi }^G:M_n(\mathbb{C})\to \mathbb{C}$ 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.