Lowering operators, orthogonal decomposition of tensor space, and quantized Schur--Weyl duality

Abstract

For $q$ generic, Jimbo showed that $q$-tensor space $V_q^{\otimes r}$ (where $V_q$ is the $n$-dimensional vector representation) satisfies Schur--Weyl duality with respect to the commuting actions of the quantized enveloping algebra $\mathbf{U}_q(\mathfrak{gl}_n)$ 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. Using a recursive construction of certain linear combinations $\Psi_j$ of Coxeter monomials in the negative part of $\mathbf{U}_q(\mathfrak{gl}_n)$, 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 (Journal of Algebra 2024) of the first two authors for the case $n =2$. Our construction works over any field containing a non-zero element $q$ which is not a root of unity. The element $\Psi_j$ depends on a weight $\lambda$ and is the ``evaluation at $\lambda$'' of a certain $q$-lowering operator $\overline{\Psi}_j$ satisfying a similar recursion, up to renormalization. This simplifies the construction of lowering operators. Both $\Psi_j$ and $\overline{\Psi}_j$ are independent of a choice of root vectors. On the other hand, the $\Psi_j$ can be applied to construct root vectors (independent of the braid group action) as explicit linear combinations of Coxeter monomials.