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two generalized matrix functions on the set of all symmetric matrices, $\\mathbb{S}_n(\\mathbb{C})$.\n  In order to fulfill the purpose, a symmetric matrix $S_\\sigma$ is constructed and $d_\\chi^G(S_\\sigma)$ is evaluated for each $\\sigma \\in S_n$.\n  By applying the value of $d_\\chi^G(S_\\sigma)$, it is shown that 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