For a finite Coxeter group~ and a Coxeter element~ of the -Cambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of~. Its maximal cones are naturally indexed by the -sortable elements of~. The main result of this paper is that the known bijection between -sortable elements and -clusters induces a combinatorial isomorphism of fans. In particular, the -Cambrian fan is combinatorially isomorphic to the normal fan of the generalized associahedron for~. The rays of the -Cambrian fan are generated by certain vectors in the -orbit of the fundamental weights, while the rays of the -cluster fan are generated by certain roots. For particular (``bipartite'') choices of~, we show that the -Cambrian fan is linearly isomorphic to the -cluster fan. We characterize, in terms of the combinatorics of clusters, the partial order induced, via the map , on -clusters by the -Cambrian lattice. We give a simple bijection from -clusters to -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 -vectors and quasi-Cartan companions.
Cite this article
Nathan Reading, David E. Speyer, Cambrian fans. J. Eur. Math. Soc. 11 (2009), no. 2, pp. 407–447DOI 10.4171/JEMS/155