On the graded identities and cocharacters of the algebra of 3$times;3 matrices*1

Abstract

Let M 2,1 ( F ) be the algebra of 3×3 matrices over an algebraically closed field F of characteristic zero with non-trivial Z 2 -grading. We study the graded identities of this algebra through the representation theory of the hyperoctahedral group Z 2 ∼S n . After splitting the space of multilinear polynomial identities into the sum of irreducibles under the Z 2 ∼S n -action, we determine all the irreducible Z 2 ∼S n -characters appearing in this decomposition with non-zero multiplicity. We then apply this result in order to study the graded cocharacter of the Grassmann envelope of M 2,1 ( F ). Finally, using the representation theory of the general linear group, we determine all the graded polynomial identities of the algebra M 2,1 ( F ) up to degree 5.