An orthogonal bimodule decomposition of quantized tensor space realizing Jimbo's Schur--Weyl duality
Stephen Doty, Anthony Giaquinto, Stuart Martin
TL;DR
The paper provides a complete, combinatorial realization of the isotypic semisimple decomposition of $V_q^{\otimes r}$ under the commuting actions of $\mathbf{U}_q(\mathfrak{gl}_n)$ and $\mathbf{H}_q(\mathfrak{S}_r)$, indexing components by walks on the Bratteli diagram. It introduces the $\Psi$ and $\Phi$ operators, built from Coxeter monomials, to construct explicit pairwise-orthogonal highest-weight vectors $\mathfrak{c}_\pi$ corresponding to each Bratteli walk $\pi$, thereby realizing the decomposition into $V_q(\lambda)\otimes \mathsf{S}_q^\lambda$ and its multiplicities. The work proves orthogonality and completeness of these vectors, yielding a hands-on basis for the isotypic components and a basis for invariants of the restricted $\mathbf{U}_q(\mathfrak{sl}_n)$-action. It also furnishes a self-contained proof of Jimbo’s Schur--Weyl duality over any field with $q\ne 0$ not a root of unity, and provides an elementary, implementable combinatorial construction that may extend to other types.
Abstract
Consider the vector representation $V_q$ of the quantized enveloping algebra $\mathbf{U}_q(\mathfrak{gl}_n)$. For $q$ generic, Jimbo showed that $q$-tensor space $V_q^{\otimes r}$ satisfies Schur--Weyl duality for the commuting actions of $\mathbf{U}_q(\mathfrak{gl}_n)$ and the Iwahori--Hecke algebra $\mathbf{H}_q(\mathfrak{S}_r)$, with the latter action derived from the $R$-matrix. In the limit as $q \to 1$, one recovers classical Schur--Weyl duality. We give a combinatorial realization of the corresponding isotypic semisimple decomposition of $V_q^{\otimes r}$ indexed by paths in the Bratteli diagram. This extends earlier work (\emph{Journal of Algebra} 2024) of the first two authors for the $n=2$ case. Our construction works over any field containing a non-zero element $q$ which is not a root of unity.