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Hook Theorem for Superalgebras with Superinvolution or Graded Involution

Irina Sviridova, Renata A. Silva

TL;DR

This work extends the classical hook theorem for polynomial identities to the setting of $\#$-superidentities in PI $\#$-superalgebras with a superinvolution or graded involution over characteristic-zero fields. It develops $\#$-cocharacters and an accompanying representation-theoretic framework for the direct product of symmetric groups, establishing a quadruple hook containment result and deriving Amitsur-type identities in this context. A key contribution is the construction of Amitsur $\#$-superpolynomials $E_{\langle d,l \rangle}$ that translate the hook data into polynomial identities, providing necessary and sufficient conditions via multiplicity vanishing outside the hook. The results yield concrete Amitsur $#$-superidentities for all PI $\#$-superalgebras and are illustrated by Grassmann-algebra examples, refining our understanding of how involution structures interact with super 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.

Hook Theorem for Superalgebras with Superinvolution or Graded Involution

TL;DR

This work extends the classical hook theorem for polynomial identities to the setting of -superidentities in PI -superalgebras with a superinvolution or graded involution over characteristic-zero fields. It develops -cocharacters and an accompanying representation-theoretic framework for the direct product of symmetric groups, establishing a quadruple hook containment result and deriving Amitsur-type identities in this context. A key contribution is the construction of Amitsur -superpolynomials that translate the hook data into polynomial identities, providing necessary and sufficient conditions via multiplicity vanishing outside the hook. The results yield concrete Amitsur -superidentities for all PI -superalgebras and are illustrated by Grassmann-algebra examples, refining our understanding of how involution structures interact with super identities.

Abstract

We consider a superalgebra with a superinvolution or graded involution over a field of characteristic zero and assume that it is a -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.
Paper Structure (7 sections, 15 theorems, 121 equations, 1 table)

This paper contains 7 sections, 15 theorems, 121 equations, 1 table.

Table of Contents

  1. Introduction
  2. Superalgebras, Superinvolutions, Graded Involutions, Identities.
  3. $FS_{\langle n \rangle}$-representations and Action of $S_{\langle n \rangle}$ in $P_{\langle n \rangle}$
  4. Main Results
  5. Amitsur's Theorem
  6. Examples
  7. Polynomials of Amitsur for $\#$-Superalgebras

Key Result

Lemma 2.2

Let $A$ be a $\#$-superalgebra. Then, for any $n \geq 1$, we have

Theorems & Definitions (33)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Proposition 3.5
  • proof
  • ...and 23 more