Crystal monoids & crystal bases: rewriting systems and biautomatic structures for plactic monoids of types An, Bn, Cn, Dn, and G2

Cain, Alan, Gray, Robert and Malheiro, António (2019) Crystal monoids & crystal bases: rewriting systems and biautomatic structures for plactic monoids of types An, Bn, Cn, Dn, and G2. Journal of Combinatorial Theory, Series A, 162. pp. 406-466. ISSN 0097-3165

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    Abstract

    The vertices of any (combinatorial) Kashiwara crystal graph carry a natural monoid structure given by identifying words labelling vertices that appear in the same position of isomorphic components of the crystal. Working on a purely combinatorial and monoid-theoretical level, we prove some foundational results for these crystal monoids, including the observation that they have decidable word problem when their weight monoid is a finite rank free abelian group. The problem of constructing finite complete rewriting systems, and biautomatic structures, for crystal monoids is then investigated. In the case of Kashiwara crystals of types An, Bn, Cn, Dn, and G2 (corresponding to the q-analogues of the Lie algebras of these types) these monoids are precisely the generalised plactic monoids investigated in work of Lecouvey. We construct presentations via finite complete rewriting systems for all of these types using a unified proof strategy that depends on Kashiwara's crystal bases and analogies of Young tableaux, and on Lecouvey's presentations for these monoids. As corollaries, we deduce that plactic monoids of these types have finite derivation type and satisfy the homological finiteness properties left and right FP∞. These rewriting systems are then applied to show that plactic monoids of these types are biautomatic and thus have word problem soluble in quadratic time.

    Item Type: Article
    Faculty \ School: Faculty of Science > School of Mathematics
    Depositing User: LivePure Connector
    Date Deposited: 08 Nov 2018 14:30
    Last Modified: 01 Feb 2019 04:30
    URI: https://ueaeprints.uea.ac.uk/id/eprint/68819
    DOI: 10.1016/j.jcta.2018.11.010

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