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.