Mixed Schur-Weyl-Sergeev duality for queer Lie superalgebras

Abstract

We introduce a new family of superalgebras ${\overrightarrow{B}}_{r,s}$ for $r,s\ge 0$ such that $r+s>0$, which we call the walled Brauer superalgebras, and prove the mixed Scur-Weyl-Sergeev duality for queer Lie superalgebras. More precisely, let $\mathfrak{q}(n)$ be the queer Lie superalgebra, $\mathbf{V}={\mathbb{C}}^{n|n}$ the natural representation of $\mathfrak{q}(n)$ and $\mathbf{W}$ the dual of $\mathbf{V}$. We prove that, if $n\ge r+s$, the superalgebra ${\overrightarrow{B}}_{r,s}$ is isomorphic to the supercentralizer algebra $_{\mathfrak{q}(n)}({\mathbf V}^{\otimes r} \otimes {\mathbf W}^{\otimes s})^{\op}$ of the $\mathfrak{q}(n)$-action on the mixed tensor space ${\mathbf{V}}^{\otimes r}\otimes {\mathbf{W}}^{\otimes s}$. As an ingredient for the proof of our main result, we construct a new diagrammatic realization ${\overrightarrow{D}}_k$ of the Sergeev superalgebra $Se{r}_k$. Finally, we give a presentation of ${\overrightarrow{B}}_{r,s}$ in terms of generators and relations.