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

TL;DR

This paper introduces a new algebraic structure called the degenerate two boundary affine Hecke-Clifford algebra, explores its actions on tensor modules of the Lie superalgebra rak{q}(n), and constructs explicit modules using combinatorial methods, establishing a Schur-Weyl type duality.

Contribution

It defines the algebra rak{H}_d, constructs explicit modules via combinatorics, and classifies a class of calibrated modules, advancing understanding of rak{q}(n) representations.

Findings

The algebra rak{H}_d admits a rak{q}(n)-linear action on tensor spaces.

Explicit modules for a quotient algebra rak{H}_{p,d} are constructed using shifted tableaux.

A weak Schur-Weyl duality is established through the decomposition of tensor modules.

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.