2018/11/06 by Isobel Webster, Webster, Isobel
Mathematics · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.1811.02257
openalex publication_date 2018/11/06 · openalex created_date 2018/11/16 · openalex updated_date 2026/07/28
Each quiver appearing in a seed of a cluster algebra determines a corresponding group, which we call a cluster group, which is 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. As for finite Coxeter groups, we can consider parabolic subgroups of cluster groups. We prove that, in the type An case, there exists an isomorphism between the lattice of subsets of the defining generators of the cluster group and the lattice of its parabolic subgroups. Moreover, each parabolic subgroup has a presentation given by restricting the presentation of the whole group.