Let $G{leq}S_n$ and ${chi}$ be any nonzero complex valued function on G. We first study the irreducibility of the generalized matrix polynomial $d^G_{chi}(X)$, where $X=(x_{ij})$ is an n-by-n matrix whose entries are $n^2$ commuting independent indeterminates over $mathbb{C}$. In particular, we show that if $mathcal{X}$ is an irreducible character of G, then $d^G_{chi}(X)$ is an irreducible polynomial, where either $G=S_n$ or $G=A_n$ and $n{ eq}2$. We then give a necessary and sufficient condition for the equality of two generalized matrix functions on the set of the so-called ${chi}$-singular (${chi}$-nonsingular) matrices.