Let $Gleq S_n$ 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 $n eq 2.$ 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.
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机译：令$ G leq S_n $和$ chi $为$ G $上的任何非零复数值函数。我们首先研究广义矩阵多项式$ d _ { chi} ^ {G}（X）$的不可约性，其中$ X =（x_ {ij}）$是一个$ n $ -by- $ n $矩阵，其项是$ n ^ 2 $个通勤独立的，在$ mathbb {C}上不确定。$特别地，我们表明，如果$ chi $是$ G的不可约性，$则$ d _ { chi} ^ {G}（ X）$是一个不可约的多项式，其中$ G = S_n $或$ G = A_n $和$ n neq 2. $然后，我们给出了满足条件的两个广义矩阵函数相等的充要条件。所谓的$ chi $-奇异（$ chi $ -nonsingular）矩阵。
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