Geometric Howe dualities of finite type

TL;DR

This paper develops a geometric framework for Howe dualities involving quantum groups of finite type, establishing new dualities and explicit decompositions using partial flag varieties and algebraic methods.

Contribution

It introduces a geometric approach to quantum Howe dualities of finite type, extending previous algebraic methods to new quantum group pairs and providing explicit multiplicity-free decompositions.

Findings

Established Howe duality between quantum general linear groups via geometric methods.

Generalized algebraic approach to $ extit{ extbf{ extit{ extbf{imath}}}}$quantum groups of type AIII/IV.

Derived explicit multiplicity-free decompositions for the dualities.

Abstract

We develop a geometric approach toward an interplay between a pair of quantum Schur algebras of arbitrary finite type. Then by Beilinson-Lusztig-MacPherson's stabilization procedure in the setting of partial flag varieties of type A (resp. type B/C), the Howe duality between a pair of quantum general linear groups (resp. a pair of $\imath$quantum groups of type AIII/IV) is established. The Howe duality for quantum general linear groups has been provided via quantum coordinate algebras in [Z02]. We also generalize this algebraic approach to $\imath$quantum groups of type AIII/IV, and prove that the quantum Howe duality derived from partial flag varieties coincides with the one constructed by quantum coordinate (co)algebras. Moreover, the explicit multiplicity-free decompositions for these Howe dualities are obtained.