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

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.