Cocharacters of Bilinear Mappings and Graded Matrices

Let M _k (F) be the algebra of k ×k matrices over a field F of characteristic 0. If G is any group, we endow M _k (F) with the elementary grading induced by the k-tuple (1,...,1,g) where g?∈?G, g ²?≠?1. Then the graded identities of M _k (F) depending only on variables of homogeneous degree g and g ^???1 are obtained by a natural translation of the identities of bilinear mappings (see Bahturin and Drensky, Linear Algebra Appl 369:95–112, 2003). Here we study such identities by means of the representation theory of the symmetric group. We act with two copies of the symmetric group on a space of multilinear graded polynomials of homogeneous degree g and g ^???1 and we find an explicit decomposition of the corresponding graded cocharacter into irreducibles.