Quantum Super Littlewood Correspondences

TL;DR

This paper explores the quantum super Littlewood theory related to superimants and supersymmetric polynomials, constructing explicit bases in the context of quantum super Schur-Weyl duality.

Contribution

It explicitly constructs basis vectors for bimodules in quantum super Schur-Weyl duality and interprets quantum super immanants through tensor representation weight spaces.

Findings

Constructed explicit basis vectors for bimodules in the quantum super setting.

Provided a new interpretation of quantum super immanants via tensor representation weights.

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 $U_q(\mathfrak{gl}_{m|n})$.