Quantum Super Littlewood Correspondences

Abstract

In this paper, we study the Littlewood theory associated with the quantum super immanants and supersymmetric polynomials, including both the super case and the quantum generalization. In the setting of quantum super Schur-Weyl duality between the quantum superalgebra $U_q({\mathfrak{gl}}_{m|n})$ and the Iwahori-Hecke algebra ${\mathcal{H}}_r$ of type A, we explicitly construct basis vectors of the $(U_q({\mathfrak{gl}}_{m|n}),{\mathcal{H}}_r)$-bimodule on the tensor product space $({\mathbb{C}}^{m|n}{)}^{\otimes r}$. Using this construction, we interpret the quantum super immanants via weight spaces of covariant tensor representations of $U_q({\mathfrak{gl}}_{m|n})$.