A Double Hall Algebra Approach to Affine Quantum Schur--Weyl Theory

TL;DR

This paper explores the structure of double Ringel-Hall algebras linked to cyclic quivers and their relation to affine quantum Schur-Weyl theory, providing new insights into their algebraic presentations and representations.

Contribution

It introduces a double Hall algebra framework for affine quantum Schur-Weyl theory, connecting various algebraic structures and proposing conjectures on their integral forms.

Findings

Drinfeld-Jimbo type presentation established

Affine quantum Schur-Weyl reciprocity demonstrated

Connections with existing algebraic structures clarified

Abstract

We investigate the structure of the double Ringel-Hall algebras associated with cyclic quivers and its connections with quantum loop algebras of $\mathfrak{gl}_n$, affine quantum Schur algebras and affine Hecke algebras. This includes their Drinfeld-Jimbo type presentation, affine quantum Schur-Weyl reciprocity, representations of affine quantum Schur algebras, and connections with various existing works by Lusztig, Varagnolo-Vasserot, Schiffmann, Hubery, Chari-Pressley, Frenkel-Mukhin, etc. We will also discuss conjectures on a realization of Beilinson-Lusztig-MacPherson type and Lusztig type integral forms for double Ringel-Hall algebras.