On the identities and cocharacters of the algebra of $3 \times 3$ matrices with orthosymplectic superinvolution

Let $M_{1,2}(F)$ be the algebra of $3 \times 3$ 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 $\mathbb{H}_n = (\mathbb{Z}_2 \times \mathbb{Z}_2) \sim S_n$. We decompose the space of multilinear $*$-identities of degree $n$ into the sum of irreducibles under the $\mathbb{H}_n$-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 $M_{1,2}(F)$ up to degree $3$.