A local characterization of quasi-crystal graphs

TL;DR

This paper introduces local axioms that characterize quasi-crystal graphs, demonstrating their closure properties and structural similarities to crystal graphs, with implications for understanding their combinatorial and algebraic properties.

Contribution

It provides a new set of local axioms for quasi-crystal graphs and proves their closure under a recent tensor product, linking them to semistandard quasi-ribbon tableaux.

Findings

Quasi-crystal graphs satisfying the axioms are closed under tensor product.

Each connected component has a unique highest weight element.

Connected components are isomorphic to quasi-crystal graphs of semistandard quasi-ribbon tableaux.

Abstract

It is provided a local characterization of quasi-crystal graphs, by presenting a set of local axioms, similar to the ones introduced by Stembridge for crystal graphs of simply-laced root systems. 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.