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

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 $\mathbf{BD}$.