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 $\ast$ over a field $F$ of characteristic zero. We study the $\ast$-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 $\ast$-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 $\ast$-polynomial identities of $M_{1,2}(F)$ up to degree $3$.