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On the Identities of the Matrices over the Grassmann Algebra

Abstract The aim of this paper is to find an (upper) bound for the multiplicities of the cocharacters for the identities of the algebra E m -the matrix algebra of order m with entries from the infinite dimensional Grassmann algebra E . We study, for this purpose, in more detail the relation between the ordinary and the graded identities of a given Z 2 -graded algebra that satisfies some specific graded identities (we call them Capelli and anti-Capelli). It turns out that for such algebras there is a pretty close link between the ordinary cocharacters and multiplicities from one hand and the graded ones from the other. The information obtained in Sections V and VI is especially useful for the cocharacters and multiplicities related to "large" diagrams. This examination makes it possible to get an upper bound for the multiplicities of the cocharacters for the identities of E m , that is very similar to the bound for the multiplicities in the case of the identities of the "ordinary" matrices over afield of characteristic zero.