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. In this paper, we present the proof of a version of the celebrated hook theorem \cite{SAR} 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.