A local characterization of $B_{2}$ regular crystals

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 $A^{(1)}_{1},G^{(1)}_{2},A^{(2)}_{2},D^{(3)}_{4}$.