An orthogonal bimodule decomposition of quantized tensor space realizing Jimbo's Schur--Weyl duality

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.