A local characterization of $B_2$ regular crystals

TL;DR

This paper extends Stembridge's local characterization of regular crystals to the case of $B_2$ and certain other non-simply-laced types, providing a new axiomatization for these structures.

Contribution

It introduces a local graph-theoretic axiomatization for $B_2$ regular crystals, broadening the understanding of crystal structures beyond simply-laced types.

Findings

Provides a local characterization for $B_2$ regular crystals.

Extends axiomatization to most finite GCMs except specific types.

Offers a framework for understanding regular crystals in non-simply-laced cases.

Abstract

Stembridge characterized regular crystals associated with a simply-laced generalized Cartan matrix (GCM) in terms of local graph-theoretic quantities. We give a similar axiomatization for $B_2$ regular crystals and thus for regular crystals associated with a finite GCM except $G_2$ and an affine GCM except $A^{(1)}_{1},G^{(1)}_{2},A^{(2)}_{2},D^{(3)}_4$.