Skew Howe duality for $U_q(\mathfrak{gl}_n)$ via quantized Clifford algebras

TL;DR

This paper extends skew Howe duality to the quantum group setting using quantized Clifford algebras, providing a new operator commutant approach and explicit module decompositions.

Contribution

It introduces a novel operator commutant framework for quantum skew Howe duality via quantized Clifford algebras, generalizing classical results.

Findings

Established a double centralizer property for $U_q(\mathfrak{gl}_n)$ and $U_q(\mathfrak{gl}_m)$

Derived a multiplicity-free decomposition of tensor products of braided exterior algebras

Parametrized irreducible modules by classical dominant weights

Abstract

We develop an operator commutant version of the First Fundamental Theorem of invariant theory for the general linear quantum group $U_q(\mathfrak{gl}_n)$ by using a double centralizer property inside a quantized Clifford algebra. In particular, we show that $U_q(\mathfrak{gl}_m)$ generates the centralizer of the $U_q(\mathfrak{gl}_n)$-action on the tensor product of braided exterior algebras $\bigwedge_q(\mathbb{C}^n)^{\otimes m}$. We obtain a multiplicity-free decomposition of the $U_q(\mathfrak{gl}_n) \otimes U_q(\mathfrak{gl}_m)$-module $\bigwedge_q(\mathbb{C}^n)^{\otimes m} \cong \bigwedge_q(\mathbb{C}^{nm})$ by computing explicit joint highest weight vectors. We find that the irreducible modules in this decomposition are parametrized by the same dominant weights as in the classical case of the well-known skew $GL_n \times GL_m$-duality. Clifford algebras are an essential feature of our work: they provide a unifying framework for classical and quantized skew Howe duality results that can be extended to include orthogonal algebras of types $\mathbf{BD}$.