Coherence for plactic monoids via rewriting theory and crystal structures

Abstract

Rewriting methods have been developed for the study of coherence for algebraic objects. This consists in starting with a convergent presentation, and expliciting a family of generating confluences to obtain a coherent presentation -- one with generators, generating relations, and generating relations between relations (syzygies). In this article we develop these ideas for a class of monoids which encode the representation theory of complex symmetrizable Kac-Moody algebras, called plactic monoids. The main tools for this are the crystal realization of plactic monoids due to Kashiwara, and a class of presentations compatible with a crystal structure, called crystal presentations. We show that the compatibility of the crystal structure with the presentation reduces certain aspects of the study of plactic monoids by rewriting theory to components of highest weight in the crystal. We thus obtain reduced versions of Newman's Lemma and Critical Pair Lemma, which are results for verifying convergence of a presentation. Further we show that the family of generating confluences of a convergent crystal presentation is entirely determined by the components of highest weight. Finally we apply these constructions to the finite convergent presentations of plactic monoids of type $A_n$, $B_n$, $C_n$, $D_n$, and $G_2$, due to Cain-Gray-Malheiro.