Quantum walled Brauer-Clifford superalgebras

Abstract

We introduce a new family of superalgebras, the quantum walled Brauer-Clifford superalgebras ${\mathsf{BC}}_{r,s}(q)$. The superalgebra ${\mathsf{BC}}_{r,s}(q)$ is a quantum deformation of the walled Brauer-Clifford superalgebra ${\mathsf{BC}}_{r,s}$ and a super version of the quantum walled Brauer algebra. We prove that ${\mathsf{BC}}_{r,s}(q)$ is the centralizer superalgebra of the action of ${\mathfrak{U}}_q(\mathfrak{q}(n))$ on the mixed tensor space ${\mathbf{V}}_q^{r,s}={\mathbf{V}}_q^{\otimes r}\otimes ({\mathbf{V}}_q^{\ast }{)}^{\otimes s}$ when $n\ge r+s$, where ${\mathbf{V}}_q=\mathbb{C}(q{)}^{(n|n)}$ is the natural representation of the quantum enveloping superalgebra ${\mathfrak{U}}_q(\mathfrak{q}(n))$ and ${\mathbf{V}}_q^{\ast }$ is its dual space. We also provide a diagrammatic realization of ${\mathsf{BC}}_{r,s}(q)$ as the $(r,s)$-bead tangle algebra ${\mathsf{BT}}_{r,s}(q)$. Finally, we define the notion of $q$-Schur superalgebras of type $\mathsf{Q}$ and establish their basic properties.

32 pages; minor corrections; the proof of Theorem 2.9 and the relations (5.13) - (5.15) are changed; to appear in Journal of Algebra