On the identities and cocharacters of the algebra of 3 × 3 matrices with orthosymplectic superinvolution

Let M1,2(F) be the algebra of 3 x 3 matrices with orthosymplectic superinvolution * over a field F of characteristic zero. We study the *-identities of this algebra through the representation theory of the group Hn= (Z2 x Z2) similar to Sn. We decompose the space of multilinear *-identities of degree n into the sum of irreducibles under the Hn-action in order to study the irreducible characters appearing in this decomposition with non-zero multiplicity. Moreover, by using the representation theory of the general linear group, we determine all the *-polynomial identities of M1,2(F) up to degree 3.