Quiver varieties and dual canonical bases

TL;DR

This paper surveys recent advances in dual canonical bases for quantum and $ extit{ extbf{imath}}$quantum groups, highlighting geometric constructions, invariance properties, and connections to double canonical bases.

Contribution

It introduces new constructions of dual canonical bases, proves their invariance under braid group actions, and links quantum groups to double canonical bases, resolving several conjectures.

Findings

Dual canonical bases exhibit positivity and invariance under braid group actions.

The geometric realization generalizes Qin's work for type ADE.

Quantum groups' dual canonical bases coincide with Berenstein-Greenstein's double canonical bases.

Abstract

We survey some recent developments on the theory of dual canonical bases for quantum groups and $\imath$quantum groups. The $\imath$quiver algebras were introduced by Wang and the first author, which are used to give two realizations of quasi-split $\imath$quantum groups of type ADE: one via the $\imath$Hall algebras and the other via the quantum Grothendieck rings of Nakajima-Keller-Scherotzke quiver varieties. The geometric construction of the $\imath$quantum groups produces their dual canonical bases with positivity, generalizing Qin's geometric realization of quantum groups of type ADE. Recently, the authors provided a new construction of the dual canonical basis in the setting of $\imath$Hall algebras, and proved that it is invariant under braid group actions, and obtained the positivity of the transition matrix coefficients from the Hall basis to the dual canonical basis. As quantum groups can be regarded as $\imath$quantum groups of diagonal type, we demonstrate that the dual canonical bases of quantum groups coincide with the double canonical bases defined by Berenstein and Greenstein, and resolve several conjectures therein.