Dual and double canonical bases of quantum groups

TL;DR

This paper proves that dual canonical bases of quantum groups coincide with Berenstein-Greenstein's double canonical bases, confirming conjectures on positivity and invariance using geometric methods involving quiver varieties.

Contribution

It establishes the equivalence of dual and double canonical bases for quantum groups, connecting algebraic and geometric constructions and settling related conjectures.

Findings

Dual canonical bases coincide with double canonical bases.

Confirmed positivity and invariance conjectures.

Unified algebraic and geometric perspective on quantum groups.

Abstract

Qin established the geometric realization of entire quantum groups via perverse sheaves, which further give rise to dual canonical bases with integral and positive structure constants for quantum groups of type ADE. In this paper, we prove that the dual canonical bases of (Drinfeld double) quantum groups coincide with Berenstein--Greenstein's double canonical bases, by reinterpreting their intricate algebraic construction via the geometry of NKS quiver varieties. This result settles several conjectures therein, including those on positivity and invariance under braid group actions.