Cambrian fans

Source 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 \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.