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Coherence for Plactic Monoids via Rewriting Theory and Crystal Structures

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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 many aspects of the study of plactic monoids via rewriting theory to computations with 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, and show that this result applies to the plactic monoids of types An, Bn, Cn, Dn, and G2, via their convergent presentations due to Cain–Gray–Malheiro (Cain et al. J. Comb. Theory Ser. A. 162, 406–466, 6).

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Acknowledgements

The author is very thankful to Stéphane Gaussent and Philippe Malbos for many informative discussions on the topic, as well as to the anonymous reviewer who read the article very carefully and gave many insightful remarks on its material and structure of exposition.

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  1. Institut Camille Jordan, University Claude Bernard Lyon 1, 21 Av. Claude Bernard, Villeurbanne, 69100, France

    Uran Meha

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Correspondence to Uran Meha.

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The research leading to the results in this paper was done by the author during his doctorate at the University Claude Bernard Lyon 1, completed in 2021 and funded by École Doctorale InfoMaths in Lyon. The author is also indebted to the Institut Henri Poincaré for their funding and hospitality during the trimester program “Representation Theory” January - April 2020, where part of this work began to take shape. The author certifies that he has no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript.

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Meha, U. Coherence for Plactic Monoids via Rewriting Theory and Crystal Structures. Algebr Represent Theor 26, 2289–2312 (2023). https://doi.org/10.1007/s10468-022-10177-8

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