Crystal Structure of Localized Quantum Unipotent Coordinate Category

TL;DR

This paper establishes that the simple objects in a localized quantum unipotent coordinate category form a crystal structure isomorphic to a cellular crystal, revealing new insights into their combinatorial and categorical properties.

Contribution

It demonstrates that the simple objects in the localized category have a crystal structure matching the cellular crystal associated with a Weyl group element.

Findings

The simple objects form a crystal structure.

The crystal is isomorphic to the cellular crystal.

The crystal graph is connected.

Abstract

A localized quantum unipotent coordinate category $\widetilde{\mathscr{C}_w}$ associated with a Weyl group element $w$ is a rigid monoidal category which is obtained by applying the localization process to a subcategory of the category of finite-dimensional graded modules of a quiver Hecke algebra. We shall show that the family of the isomorphism classes (up to grading shifts) of simple objects in $\widetilde{\mathscr{C}_w}$ possesses a crystal structure and it is isomorphic to the cellular crystal associated with $w$. As an application of this result, we shall show the connectedness of the crystal graph of an arbitrary cellular crystal.