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

Abstract

The goal of our work is to study the decomposition of the joint action of $\mathfrak{g} = \mathfrak{spo}(2n|1)$ and $\mathfrak{g}' = \mathfrak{osp}(2|2)$ on the supersymmetric algebra S = 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 $\mathfrak{g}$ and $\mathfrak{g}'$ appearing as subrepresentations of S. In this paper, we obtained an explicit description of the highest weights and joint highest weight vectors for the representations of $\mathfrak{g}$ and $\mathfrak{g}'$ appearing in the duality.

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