Some results on the higher-rank graphs associated to crystals of semisimple Lie algebras

Abstract

In this paper we continue the study of the higher-rank graphs associated to finite-dimensional complex semisimple Lie algebras, introduced by the author and R. Yuncken, whose construction relies on Kashiwara's theory of crystals. First we prove that the Bruhat graphs of the corresponding Weyl groups, both weak and strong, can be embedded into the higher-rank graphs as colored graphs. Next, specializing to Lie algebras of type $A$, we connect some aspects of the construction of the higher-rank graphs with some well-known notions in combinatorics, most notably the keys of Lascoux and Sch\"utzenberger.