Hook Theorem for Superalgebras with Superinvolution or Graded Involution

TL;DR

This paper extends the hook theorem to superalgebras with superinvolution or graded involution, providing combinatorial insights into their identities and establishing analogues of Amitsur identities.

Contribution

It proves a version of the hook theorem for multilinear superidentities in superalgebras with superinvolution, and introduces Amitsur identities for these structures.

Findings

Hook theorem for superalgebras with superinvolution established

Analogues of Amitsur identities derived for superalgebras

Provides combinatorial and polynomial characterizations of identities

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.