Coherent presentations of Artin monoids

By S. Gaussent, Yves Guiraud, P. Malbos. Compositio Mathematica, 2012

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We compute coherent presentations of Artin monoids, that is, presentations by generators, relations, and relations between the relations. For that, we use methods of higher-dimensional rewriting that extend Squier’s and Knuth–Bendix’s completions into a homotopical completion–reduction, applied to Artin’s and Garside’s presentations. The main result of the paper states that the so-called Tits–Zamolodchikov 3-cells extend Artin’s presentation into a coherent presentation. As a byproduct, we give a new constructive proof of a theorem of Deligne on the actions of an Artin monoid on a category.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fa04deabff6bb765a6d2c85252eb4fbd5f090c62