GENERALIZED MATRIX FUNCTIONS, IRREDUCIBILITY AND EQUALITY

Abstract

Abstract. Let G ≤ S n and χ be any nonzero complex valued functionon G. We first study the irreducibility of the generalized matrix polyno-mial d Gχ (X), where X = (x ij ) is an n-by-n matrix whose entries are n 2 commuting independent indeterminates over C. In particular, we showthat if χ is an irreducible character of G, then d Gχ (X) is an irreduciblepolynomial, where either G = S n or G = A n and n 6= 2 . We then givea necessary and sufficient condition for the equality of two generalizedmatrix functions on the set of the so-called χ-singular (χ-nonsingular)matrices. 1. IntroductionLet S n be the symmetric group of degree n, Gan arbitrary subgroup ofS n , and let χ: G→ Cbe a function. Denote by M n (C) the set of all n-by-nmatrices over Cand define the function d Gχ : M n (C) → Cas follows:d Gχ (A) =X σ∈G χ(σ)Y ni=1 a iσ(i) ,where A= (a ij ) ∈ M n (C). The function d Gχ is called the generalized matrixfunction associated with Gand χ. Note that if G= S n and χ= 1 G is theprincipal character of G, then d