Cambrian fans

Reading, Nathan, and Speyer, David E.. "Cambrian fans." Journal of the European Mathematical Society 011.2 (2009): 407-447. <http://eudml.org/doc/277235>.

@article{Reading2009,
abstract = {For a finite Coxeter group $W$ and a Coxeter element $c$ of $W$; the $c$-Cambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of $W$. Its maximal cones are naturally indexed by the $c$-sortable elements of $W$. The main result of this paper is that the known bijection cl$_c$ between $c$-sortable elements and $c$-clusters induces a combinatorial isomorphism of fans. In particular, the $c$-Cambrian fan is combinatorially isomorphic to the normal fan of the generalized associahedron for $W$. The rays of the $c$-Cambrian fan are generated by certain vectors in the $W$-orbit of the fundamental weights, while the rays of the $c$-cluster fan are generated by certain roots. For particular (“bipartite”) choices of $c$, we show that the $c$-Cambrian fan is linearly isomorphic to the $c$-cluster fan. We characterize, in terms of the combinatorics of clusters, the partial order induced, via the map cl$_c$, on $c$-clusters by the $c$-Cambrian lattice. We give a simple bijection from $c$-clusters to $c$-noncrossing partitions that respects the refined (Narayana) enumeration. We relate the Cambrian fan to well-known objects in the theory of cluster algebras, providing a geometric context for g-vectors and quasi-Cartan companions.},
author = {Reading, Nathan, Speyer, David E.},
journal = {Journal of the European Mathematical Society},
keywords = {$c$-sortable elements; finite Coxeter groups; combinatorial isomorphisms of fans; maximal cells; Cambrian lattices; generalized associahedra; cluster algebras; -sortable elements; finite Coxeter groups; combinatorial isomorphisms of fans; maximal cells; Cambrian lattices; generalized associahedra; cluster algebras},
language = {eng},
number = {2},
pages = {407-447},
publisher = {European Mathematical Society Publishing House},
title = {Cambrian fans},
url = {http://eudml.org/doc/277235},
volume = {011},
year = {2009},
}

TY - JOUR
AU - Reading, Nathan
AU - Speyer, David E.
TI - Cambrian fans
JO - Journal of the European Mathematical Society
PY - 2009
PB - European Mathematical Society Publishing House
VL - 011
IS - 2
SP - 407
EP - 447
AB - For a finite Coxeter group $W$ and a Coxeter element $c$ of $W$; the $c$-Cambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of $W$. Its maximal cones are naturally indexed by the $c$-sortable elements of $W$. The main result of this paper is that the known bijection cl$_c$ between $c$-sortable elements and $c$-clusters induces a combinatorial isomorphism of fans. In particular, the $c$-Cambrian fan is combinatorially isomorphic to the normal fan of the generalized associahedron for $W$. The rays of the $c$-Cambrian fan are generated by certain vectors in the $W$-orbit of the fundamental weights, while the rays of the $c$-cluster fan are generated by certain roots. For particular (“bipartite”) choices of $c$, we show that the $c$-Cambrian fan is linearly isomorphic to the $c$-cluster fan. We characterize, in terms of the combinatorics of clusters, the partial order induced, via the map cl$_c$, on $c$-clusters by the $c$-Cambrian lattice. We give a simple bijection from $c$-clusters to $c$-noncrossing partitions that respects the refined (Narayana) enumeration. We relate the Cambrian fan to well-known objects in the theory of cluster algebras, providing a geometric context for g-vectors and quasi-Cartan companions.
LA - eng
KW - $c$-sortable elements; finite Coxeter groups; combinatorial isomorphisms of fans; maximal cells; Cambrian lattices; generalized associahedra; cluster algebras; -sortable elements; finite Coxeter groups; combinatorial isomorphisms of fans; maximal cells; Cambrian lattices; generalized associahedra; cluster algebras
UR - http://eudml.org/doc/277235
ER -