Mathematics – Combinatorics
Scientific paper
2007-09-27
S\'em. Lothar. Combin. 61A (2009), Art. B61Aa, 13 pp.
Mathematics
Combinatorics
12 pages, 4 figures, pdflatex: v2: correction of typos
Scientific paper
Let $(W,S)$ be a finite Coxeter system acting by reflections on an $\mathbb R$-Euclidean space with simple roots $\Delta=\{\a_s | s\in S\}$ of the same length and fundamental weights $\Delta^*=\{v_s | s\in S\}$. 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 $Asso_c(W)$ 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 $\mu(c') = c$ or $\mu(c')=w_0c^{-1}w_0$ 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 $W$.
Bergeron Nantel
Hohlweg Christophe
Lange Carsten
Thomas Helmuth
No associations
LandOfFree
Isometry classes of generalized associahedra does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Isometry classes of generalized associahedra, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Isometry classes of generalized associahedra will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-482027