Skew Howe duality for Types $\mathbf{BD}$ via $q$-Clifford algebras

TL;DR

This paper extends skew Howe duality to orthogonal types using $q$-Clifford algebras, providing a new operator commutant approach and explicit tensor decomposition results for quantum orthogonal groups.

Contribution

It introduces a novel operator commutant framework for $U_q(rak{so}_n)$ invariant theory and extends skew Howe duality to types $f B$ and $f D$ using $q$-Clifford algebras.

Findings

Achieved a multiplicity-free tensor decomposition for $U_q(rak{so}_{2n})$ spin representations.

Developed a double centralizer property within a quantized Clifford algebra.

Extended classical skew Howe duality results to orthogonal types $f B$ and $f D$.

Abstract

We extend a quantized skew Howe duality result for Type $\mathbf{A}$ algebras to orthogonal types via a seesaw. We develop an operator commutant version of the First Fundamental Theorem of invariant theory for $U_q(\mathfrak{so}_n)$ using a double centralizer property inside a quantized Clifford algebra. We obtain a multiplicity-free decomposition of tensor powers of the $U_q(\mathfrak{so}_{2n})$ spin representation by explicitly computing joint highest weights with respect to an action of $U_q(\mathfrak{so}_{2n}) \otimes U_q'(\mathfrak{so}_m)$. 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 $\mathbf{BD}$.