arXiv subjects

Uran Meha

Publications and source records attributed to Uran Meha.

2 recordsLinked to original sources

arXiv2021

Coherence for plactic monoids via rewriting theory and crystal structures

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.

math.RT↗

arXiv2020

C-trees and a coherent presentation for the plactic monoid of type C

In this article we introduce the $\mathbb{N}-$decorated plactic monoid of type $C$, denoted $Pl^\mathbb{N}(C_n)$, via a finite convergent presentation $ACol$, with generating set $ACol(C_n)$ consisting of admissible columns, and an element $ε$. By Squier's coherent completion theorem, this presentation is extended into a coherent presentation by identifying a family of generating confluences, i.e. generating $3-$cells. Here the generating $3-$cells are critical branchings on words of length $3$. We adapt the notions of crystal structure to $ACol(C_n)^\ast$ , and show that the shape of $3-$cells is preserved by the action of Kashiwara operators. Thus we reduce the study of the coherent presentation to only describing the generating $3-$cells whose source is a word of highest weight. We then introduce combinatorial objects called $C-$trees which parameterize the words of highest weight in $ACol(C_n)^\ast$. The $C-$trees allow for simplifying calculations with the insertion algorithm in type $C$, as introduced in by Lecouvey, and we prove that the generating $3-$cells in $ACol$ are of shape at most $(4,3)$. As a consequence, we show that the column presentation of $Pl(C_n)$, as introduced by Hage, has generating $3-$cells of shape at most $(4,3)$. This contrasts the situation in type $A$, where the $3-$cells in the column presentation of $Pl(A_n)$ are of shape at most $(3,3)$.

math.CO↗