Generalized matrix functions, irreducibility and equality

چکیده مقاله

Let G ≤ Sn and Χ be any nonzero complex valued function on G. We first study the irreducibility of the generalized matrix polynomial d_G^Χ (X), where X = (x_ij ) is an n-by-n matrix whose entries are n^2 commuting independent indeterminates over C. In particular, we show that if Χ is an irreducible character of G, then d_G^Χ (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 Χ-singular (Χ-nonsingular) matrices.