A local characterization of quasi-crystal graphs

Abstract

A local characterization of quasi-crystal graphs of type $A_{n-1}$ is provided, by presenting a set of local axioms, similar to the ones introduced by Stembridge for crystal graphs of simply-laced root systems, but restricted to type $A_{n-1}$. It is also shown that quasi-crystal graphs satisfying these axioms are closed under the tensor product recently introduced by Cain, Guilherme and Malheiro. It is deduced that each connected component of such a graph has a unique highest weight element, whose weight is a composition, and it is isomorphic to a quasi-crystal graph of semistandard quasi-ribbon tableaux.