Cluster groups: groups with presentations arising from cluster algebras

Each quiver appearing in a seed of a cluster algebra determines a corresponding group, which we call a cluster group, defined via a presentation. Grant and Marsh showed that, for quivers appearing in seeds of cluster algebras of finite type, the associated cluster groups are isomorphic to finite reflection groups, thus are finite Coxeter groups. There are many well-established results for Coxeter presentations and we are interested in whether cluster group presentations possess comparable properties. As for finite Coxeter groups, we can consider parabolic subgroups of cluster groups. We prove that, in the type A case, an isomorphism exists between the lattice of subsets of the set of defining generators of the cluster group and the lattice of its parabolic subgroups. Moreover, we show each parabolic subgroup has a presentation given by restricting the presentation of the whole group. In addition, we provide a method for obtaining a positive companion basis of the quiver. For more general quivers, we prove an alternative exchange lemma for the associated cluster group by showing that each element has a factorisation with respect to a given parabolic subgroup. In the type A case, we also consider elements whose reduced expressions all begin with a certain fixed generator and examine the form of these reduced expressions. Finally, we provide an alternative proof to the known fact that the length function on a parabolic subgroup of a Coxeter group of type A agrees with the length function on the whole group and discuss an analogous conjecture for cluster groups.