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}$$isafunctionconstructedbyasubgroup$G$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 $d_\chi^G(AB) = d_\chi^G(A)d_\chi^G(B)$foreach$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.