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

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 $U_q({\mathfrak{so}}_{2n})\otimes U_q^{\prime }({\mathfrak{so}}_m)$. 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 $\mathbf{BD}$.