On the local structure of doubly laced crystalsAlgebra & Discrete Mathematics
|Philip Sternberg, UC Davis
|Thu, Apr 6 2006, 12:10PM
Crystal bases, (or simply crystals) are colored directed graphs with representation theoretic meaning. They were originally defined by finding a distinguished basis of a quotient of a quantum group module; since then many combinatorial characterizations of crystals have been given, all but one of them global in nature.
Stembridge (2003) provided a list of eight axioms, each of which addresses local properties of a graph, that characterizes crystal graphs for representations of simply laced algebras. He also conjectures a prescription for the local behavior of crystals for representations of doubly laced algebras. We confirm this conjecture, and discuss a possible approach to providing a characterization of these crystals.