On crystal operators in Lusztig's parametrizations and string cone defining inequalities

TL;DR

This paper compares Lusztig's and Kashiwara's parametrizations of the canonical basis for type A quivers, revealing how crystal operators define inequalities of the string cone via combinatorics of Auslander-Reiten quivers.

Contribution

It establishes that Lusztig's parametrization patterns correspond to the defining inequalities of the string cone, providing a combinatorial approach using Auslander-Reiten quivers.

Findings

Crystal operators act in finitely many patterns in Lusztig's parametrization.

The set of patterns corresponds to the inequalities of the string cone.

Provides an alternative enumeration of the string cone inequalities using hammocks.

Abstract

Let $\mathbf{w}_0$ be a reduced expression for the longest element of the Weyl group, adapted to a quiver of type $A_n$. We compare Lusztig's and Kashiwara's (string) parametrizations of the canonical basis associated with $\mathbf{w}_0$. Crystal operators act in a finite number of patterns in Lusztig's parametrization, which may be seen as vectors. We show this set gives the system of defining inequalities of the string cone constructed by Gleizer and Postnikov. We use combinatorics of Auslander-Reiten quivers, and as a by-product we get an alternative enumeration of a set of inequalities defining the string cone, based on hammocks.