Artin Group Presentations Arising from Cluster Algebras

TL;DR

This paper extends the connection between cluster algebras and Artin groups by defining new presentations that are invariant under mutations, enriching the algebraic understanding of these structures.

Contribution

It introduces mutation-invariant Artin group presentations derived from cluster algebra diagrams, generalizing previous reflection group results.

Findings

Defined new generator relations for Artin groups

Proved mutation-invariance of these Artin group presentations

Extended prior results from reflection groups to Artin groups

Abstract

In 2003, Fomin and Zelevinsky proved that finite type cluster algebras can be classified by Dynkin diagrams. Then in 2013, Barot and Marsh defined the presentation of a reflection group associated to a Dynkin diagram in terms of an edge-weighted, oriented graph, and proved that this group is invariant (up to isomorphism) under diagram mutations. In this paper, we extend Barot and Marsh's results to Artin group presentations, defining new generator relations and showing mutation-invariance for these presentations.