Howe duality for the dual pair $\left(\text{SpO}(2n|1)\,, \mathfrak{osp}(2|2)\right)$

TL;DR

This paper investigates Howe duality between the supergroups SpO(2n|1) and the Lie superalgebra osp(2|2), providing explicit descriptions of their joint representations and highest weights in the supersymmetric algebra setting.

Contribution

It offers a detailed explicit characterization of the highest weights and joint highest weight vectors for the dual pair's representations.

Findings

Explicit highest weight descriptions for dual pair representations

Identification of joint highest weight vectors

Clarification of the duality correspondence in supersymmetric algebra

Abstract

The goal of our work is to study the decomposition of the joint action of $\mathscr{G} = \text{SpO}(2n|1)$ and $\mathfrak{g}' = \mathfrak{osp}(2|2)$ on the supersymmetric algebra $\text{S} = \text{S}(\mathbb{C}^{2n|1} \otimes \mathbb{C}^{1|1})$. As proved by Merino and Salmasian, we have a one-to-one correspondence between irreducible representations of $\mathscr{G}$ and $\mathfrak{g}'$ appearing as subrepresentations of $\text{S}$. In this paper, we obtained an explicit description of the highest weights and joint highest weight vectors for the representations of $\mathscr{G}$ and $\mathfrak{g}'$ appearing in the duality.