Automorphisms of the generalized cluster complex
Matthieu Josuat-Vergès
TL;DR
This paper determines the full automorphism structure of the generalized cluster complex $Γ^{(m)}$ associated to a finite-type Coxeter system. It shows that a dihedral symmetry generated by the maps $R$ and $S$ (or $R$ and $T$), together with diagram automorphisms, almost determines $Aut(Γ^{(m)})$, yielding $Aut(Γ^{(m)})=Dih\rtimes(Diag/⟨C⟩)$ and the explicit order $|Aut(Γ^{(m)})|=(mh+2)ω$. The analysis hinges on the compatibility relation, a reflection ordering, and a detailed study of links and parabolic subcomplexes, plus the introduction of the involutive automorphisms $\mathcal{S}$ and $\mathcal{T}$. These results generalize the classical type A picture, provide a robust framework for symmetries in generalized cluster combinatorics, and point toward connections with cluster parking functions and representation theory.
Abstract
It is proved that the generalized cluster complex defined by Fomin and Reading has a dihedral symmetry. Together with diagram symmetries, they generate its automorphism group. A consequence is a simple explicit formula for the order of this automorphism group.