Two Boundary Centralizer Algebras for $\mathfrak{q}(n)$

Abstract

We define the degenerate two boundary affine Hecke-Clifford algebra ${\mathcal{H}}_d$, and show it admits a well-defined $\mathfrak{q}(n)$-linear action on the tensor space $M\otimes N\otimes V^{\otimes d}$, where $V$ is the natural module for $\mathfrak{q}(n)$, and $M,N$ are arbitrary modules for $\mathfrak{q}(n)$, the Lie superalgebra of Type Q. When $M$ and $N$ are irreducible highest weight modules parameterized by a staircase partition and a single row, respectively, this action factors through a quotient of ${\mathcal{H}}_d$. We then construct explicit modules for this quotient, ${\mathcal{H}}_{p,d}$, using combinatorial tools such as shifted tableaux and the Bratteli graph. These modules belong to a family of modules which we call calibrated. Using the relations in ${\mathcal{H}}_{p,d}$, we also classify a specific class of calibrated modules. The irreducible summands of $M\otimes N\otimes V^{\otimes d}$ coincide with the combinatorial construction, and provide a weak version of the Schur-Weyl type duality.