Coxeter groups and their quotients arising from cluster algebras

TL;DR

This paper extends the explicit construction of Coxeter group presentations from cluster algebra seeds to affine cases and diagrams from surfaces and orbifolds, providing new quotient group descriptions.

Contribution

It generalizes previous constructions to affine Coxeter groups and diagrams from surfaces and orbifolds, offering new presentations as quotients of Coxeter groups.

Findings

Presentations for all affine Coxeter groups derived from cluster algebra seeds.

New quotient group descriptions for diagrams from surfaces and orbifolds.

Extension of previous finite Weyl group constructions to affine and geometric cases.

Abstract

In a recent paper, Barot and Marsh presented an explicit construction of presentation of a finite Weyl group by any seed of corresponding cluster algebra, i.e. by any diagram mutation-equivalent to an orientation of a Dynkin diagram with given Weyl group. Extending their construction to the affine case, we obtain presentations for all affine Coxeter groups. Furthermore, we generalize the construction to the settings of diagrams arising from unpunctured triangulated surfaces and orbifolds, which leads to presentations of corresponding groups as quotients of numerous distinct Coxeter groups.