On the graded identities and cocharacters of the algebra of 3 × 3 matrices

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.