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.