Specht property for the $2$-graded identities of $B_m$

TL;DR

This paper proves that the ideal of all 2-graded identities of the Jordan superalgebra $B_m$ has the Specht property and computes its 2-graded cocharacter sequence, advancing understanding of polynomial identities in superalgebras.

Contribution

It establishes the Specht property for the 2-graded identities of $B_m$ and explicitly computes its 2-graded cocharacter sequence, a novel result in superalgebra theory.

Findings

The ideal of 2-graded identities of $B_m$ satisfies the Specht property.

The 2-graded cocharacter sequence of $B_m$ has been explicitly computed.

The results deepen the understanding of polynomial identities in Jordan superalgebras.

Abstract

Let $K$ be a field of characteristic zero and $V$ a vector space of dimension $m>1$ with a nondegenerate symmetric bilinear form $f:V\times V \rightarrow K$. The Jordan algebra $B_m=K\oplus V$ of the form $f$ is a superalgebra with this decomposition. We prove that the ideal of all the $2$-graded identities of $B_m$ satisfies the Specht property and we compute the $2$-graded cocharacter sequence of $B_m$.