Isometry classes of generalized associahedra

Let $(W,S)$ be a finite Coxeter system acting by reflections on an $\mathbb R$-Euclidean space with simple roots $Δ=\{\a_s | s\in S\}$ of the same length and fundamental weights $Δ^*=\{v_s | s\in S\}$. We set $M(e)=\sum_{s\in S}κ_s v_s$, $κ_s>0$, and for $w\in W$ we set $M(w)=w(M(e))$. The permutahedron $Perm(W)$ is the convex hull of the set $\{M(w) | w\in W\}$. Given a Coxeter element $c\in W$, we have defined in a previous work a generalized associahedron $Asso_c(W)$ whose normal fan is the corresponding $c$-Cambrian fan $F_c$ defined by N. Reading. By construction, $Asso_c(W)$ is obtained from $Perm(W)$ by removing some halfspaces according to a rule prescribed by $c$. In this work, we classify the isometry classes of these realizations. More precisely, for $(W,S)$ an irreducible finite Coxeter system and $c,c'$ two Coxeter elements in $W$, we have that $Asso_{c}(W)$ and $Asso_{c'}(W)$ are isometric if and only if $μ(c') = c$ or $μ(c')=w_0c^{-1}w_0$ for $μ$ an automorphism of the Coxeter graph of $W$ such that $κ_s=κ_{μ(s)}$ for all $s\in S$. As a byproduct, we classify the isometric Cambrian fans of $W$.

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