Bulletin of the
Korean Mathematical Society BKMS

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 $d_{\chi}^{G}(X)$, 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 $\chi$ is an irreducible character of $G,$ then $d_{\chi}^{G}(X)$ is an irreducible polynomial, where either $G=S_n$ or $G=A_n$ and $n \neq 2.$ 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.

Keywords: generalized matrix functions, irreducibility, $\chi$-singular and $\chi$-nonsingular matrices

MSC numbers: 15A15, 12E05, 20C15