Howe duality for the dual pair $(\text{SpO}(2n\mathrm{\mid }1),\mathfrak{o}\mathfrak{s}\mathfrak{p}(2\mathrm{\mid }2))$

The goal of our work is to study the decomposition of the joint action of $\mathcal{G}=\text{SpO}(2n\mathrm{\mid }1)$ and ${\mathfrak{g}}^{\mathrm{\prime }}=\mathfrak{o}\mathfrak{s}\mathfrak{p}(2\mathrm{\mid }2)$ on the supersymmetric algebra $\text{S}=\text{S}({\mathbb{C}}^{2n\mathrm{\mid }1}\otimes {\mathbb{C}}^{1\mathrm{\mid }1})$. As proved by Merino and Salmasian, we have a one-to-one correspondence between irreducible representations of $\mathcal{G}$ and ${\mathfrak{g}}^{\mathrm{\prime }}$ 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 $\mathcal{G}$ and ${\mathfrak{g}}^{\mathrm{\prime }}$ appearing in the duality.