Dual Braid Presentations and Cluster Algebras

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2022-08-01T05:00:00.000Z

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Abstract

Presentations for Coxeter groups and their braid groups are encoded by Dynkin diagrams. In their foundational work on cluster algebras, Fomin and Zelevinsky defined an operation on quivers (oriented Dynkin diagrams) called mutation. It is reasonable to ask if a quiver mutation-equivalent to (an orientation of) a Dynkin diagram also encodes a presentation of a Coxeter or braid group. By explicitly writing down a set of relations, Barot and Marsh constructed such presentations for Coxeter groups, which Grant and Marsh generalized to the corresponding braid groups. We explain and generalize these results for simply-laced types using presentations encoded by reduced factorizations (into reflections) of a Coxeter element—the results above are recovered by specializing to certain two-part factorizations (in bijection with vertices of the cluster exchange graph) and certain compositions of Hurwitz moves (paralleling quiver mutation).

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