
arXiv: math/0606201
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 \operatorname{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 \operatorname{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 \mathbf{g} -vectors and quasi-Cartan companions.
cluster algebras, maximal cells, Primary 20F55; Secondary 16S99, \(c\)-sortable elements, combinatorial isomorphisms of fans, Cambrian lattices, Cluster algebras, generalized associahedra, Arrangements of points, flats, hyperplanes (aspects of discrete geometry), Combinatorial aspects of groups and algebras, Combinatorics of partially ordered sets, finite Coxeter groups, Secondary 16S99, Reflection and Coxeter groups (group-theoretic aspects), Primary 20F55, FOS: Mathematics, Mathematics - Combinatorics, Combinatorics (math.CO)
cluster algebras, maximal cells, Primary 20F55; Secondary 16S99, \(c\)-sortable elements, combinatorial isomorphisms of fans, Cambrian lattices, Cluster algebras, generalized associahedra, Arrangements of points, flats, hyperplanes (aspects of discrete geometry), Combinatorial aspects of groups and algebras, Combinatorics of partially ordered sets, finite Coxeter groups, Secondary 16S99, Reflection and Coxeter groups (group-theoretic aspects), Primary 20F55, FOS: Mathematics, Mathematics - Combinatorics, Combinatorics (math.CO)
| selected citations These citations are derived from selected sources. This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically). | 38 | |
| popularity This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network. | Top 10% | |
| influence This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically). | Top 10% | |
| impulse This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network. | Top 10% |
