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 $\hat{I}{¨}_j$ 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 $\hat{I}{¨}_j$ depends on a weight $\hat{I}»$ and is the ``evaluation at $\hat{I}»$'' of a certain $q$-lowering operator $\overline{\hat{I}}{¨}_j$ satisfying a similar recursion, up to renormalization. This simplifies the construction of lowering operators. Both $\hat{I}{¨}_j$ and $\overline{\hat{I}}{¨}_j$ are independent of a choice of root vectors. On the other hand, the $\hat{I}{¨}_j$ can be applied to construct root vectors (independent of the braid group action) as explicit linear combinations of Coxeter monomials.

34 pages