On the graded identities and cocharacters of the algebra of 3×3 matrices Original Research Article

Let M2,1(F) be the algebra of 3×3 matrices over an algebraically closed field F of characteristic zero with non-trivial image-grading. We study the graded identities of this algebra through the representation theory of the hyperoctahedral group image. After splitting the space of multilinear polynomial identities into the sum of irreducibles under the image-action, we determine all the irreducible image-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 M2,1(F). Finally, using the representation theory of the general linear group, we determine all the graded polynomial identities of the algebra M2,1(F) up to degree 5.