Hook theorem for superalgebras with superinvolution or graded involution

Abstract

We consider a superalgebra with a superinvolution or graded involution # over a field F of characteristic zero and assume that it is a PI-algebra. S.A. Amitsur and A. Regev have proved in 1982 the celebrated hook theorem for ordinary polynomial identities. In this paper, we present the proof of a version of the hook theorem for the case of multilinear #-superidentities. This theorem provides important combinatorial characteristics of identities in the language of symmetric group representations. Furthermore, we present an analogue of Amitsur identities for #-superalgebras, which are polynomial interpretations of the mentioned combinatorial characteristics, as a consequence of the hook theorem.