Isometry classes of generalized associahedra

Abstract

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 ${\Delta }^{\ast }=\{v_s|s\in S\}$. We set $M(e)={\sum }_{s\in S}{\kappa }_s{v}_s$, ${\kappa }_s>0$, 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 $Ass{o}_c(W)$ whose normal fan is the corresponding $c$-Cambrian fan $F_c$ defined by N. Reading. By construction, $Ass{o}_c(W)$ 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 $c,c^{\prime }$ two Coxeter elements in $W$, we have that $Ass{o}_c(W)$ and $Ass{o}_{c^{\prime }}(W)$ are isometric if and only if $\mu (c^{\prime })=c$ or $\mu (c^{\prime })=w_0{c}^{-1}{w}_0$ for $\mu$ an automorphism of the Coxeter graph of $W$ such that ${\kappa }_s={\kappa }_{\mu (s)}$ for all $s\in S$. As a byproduct, we classify the isometric Cambrian fans of $W$.

12 pages, 4 figures, pdflatex: v2: correction of typos

References in corpus (2)

• Cambrian fans
• Sortable elements and Cambrian lattices

Cited by in corpus (6)

• The brick polytope of a sorting network
• Compatibility fans for graphical nested complexes
• Minkowski Decomposition of Associahedra and Related Combinatorics
• Permutahedra and Associahedra: Generalized associahedra from the geometry of finite reflection groups
• Signed tree associahedra
• Permutahedra and generalized associahedra