Standard polynomials and matrices with superinvolutions

Let Mn(F) be the algebra of n×n matrices over a field F of characteristic zero. The superinvolutions ⁎ on Mn(F) were classified by Racine in [12]. They are of two types, the transpose and the orthosymplectic superinvolution. This paper is devoted to the study of ⁎-polynomial identities satisfied by Mn(F). The goal is twofold. On one hand, we determine the minimal degree of a standard polynomial vanishing on suitable subsets of symmetric or skew-symmetric matrices for both types of superinvolutions. On the other, in case of M2(F), we find generators of the ideal of ⁎-identities and we compute the corresponding sequences of cocharacters and codimensions.