# Cambrian fans

### Nathan Reading

University of Michigan, Ann Arbor, United States### David E. Speyer

Massachusetts Institute of Technology, Cambridge, United States

## 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 $\mathbf{g}$-vectors and quasi-Cartan companions.

## Cite this article

Nathan Reading, David E. Speyer, Cambrian fans. J. Eur. Math. Soc. 11 (2009), no. 2, pp. 407–447

DOI 10.4171/JEMS/155