Cambrian fans

Nathan Reading ; David E. Speyer

Journal of the European Mathematical Society, Tome 11 (2009) no. 2, pp. 407-447

Résumé

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 clc​ 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 clc​, 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 g-vectors and quasi-Cartan companions.

Détail

DOI : 10.4171/jems/155

Classification : 20-XX, 16-XX, 00-XX