# Cambrian fans

Nathan Reading; David E. Speyer

Journal of the European Mathematical Society (2009)

- Volume: 011, Issue: 2, page 407-447
- ISSN: 1435-9855

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topReading, 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 -

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