Geometric Schur Duality of two parameter quantum group of type A

TL;DR

This paper provides a geometric framework for understanding Schur-Weyl duality in two-parameter quantum groups of type A, extending classical duality results and establishing new dualities with Hecke algebras.

Contribution

It introduces a geometric description of Schur-Weyl duality for two-parameter quantum algebras and establishes dualities with Hecke algebras, extending classical results.

Findings

Geometric description of Schur-Weyl duality for $U_{v,t}(gl_n)$.

Reduction of classical duality to two-parameter case via Galois descent.

New dualities between quantum algebras and Hecke algebras.

Abstract

In this paper, we give an geometric description of the Schur-Weyl duality for two-parameter quantum algebras $U_{v, t}(gl_n)$, where $U_{v, t}(gl_n)$ is the deformation of $U_v(I, \cdot)$, the classic Shur-Weyl duality $(U_{r, s}(gl_n), V^{\otimes d}, H_d(r, s))$ can be seen as a corollary of the Shur-Weyl duality $(U_{v, t}(gl_n), V^{\otimes d}, H_d(v, t))$ by using the galois descend approach. we also establish the Shur-Weyl duality between the algebras $\widetilde{U_{v, t}(gl_N)^m}$, $\widehat{U_{v, t}(gl_N)^m}$ and Heck algebra $H_k(v, t)$.