On the graded identities and cocharacters of the algebra of \(3\times 3\) matrices (Q1827478)

In this paper, the graded identities of the algebra of \(3\times 3\) matrices (over an algebraically closed field of characteristic zero with non-trivial \(\mathbb Z_2\)-grading) through the representation theory of the hyperoctahedral group \(\mathbb Z_2-S_n\) are studied. By decomposing the space of multilinear polynomial identities into the sum of irreducibles under the \(\mathbb Z_2-S_n\)-action, all the irreducible \(\mathbb Z_2-S_n\)-characters of this decomposition with non-zero multiplicity can be determined. This allows studying the graded cocharacter of the Grassmann envelope of the mentioned algebra. Further, all the graded polynomial identities of this algebra up to degree 5 are determined via the representation theory of the general linear group.