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

Abstract

We define the degenerate two boundary affine Hecke-Clifford algebra H d , and show it admits a well-defined q(n)-linear action on the tensor space M ⊗ N ⊗ V ⊗d , where V is the natural module for q(n), and M, N are arbitrary modules for 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 H d . We then construct explicit modules for this quotient, 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 H p,d , we also classify a specific class of calibrated modules. The irreducible summands of M ⊗ N ⊗ V ⊗d coincide with the combinatorial construction, and provide a weak version of the Schur-Weyl type duality.