ON THE IDENTITIES OF THE MATRICES OVER THE GRASSMANN ALGEBRA

The aim of this paper is to find an (upper) bound for the multipliciti es of the cocharacters for the identities of the algebra E(m)-the matr ix algebra of order m with entries from the infinite dimensional Grass mann algebra E. We study, for this purpose, in more detail the relatio n between the ordinary and the graded identities of a given Z(2)-grade d 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 multiplic ities from one hand and the graded ones from the other. The informatio n obtained in Sections V and VI is especially useful for the cocharact ers 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, that is very similar to the bou nd for the multiplicities in the case of the identities of the ''ordin ary'' matrices over afield of characteristic zero. (C) 1994 Academic P ress, Inc.