JournalsjemsVol. 11, No. 2pp. 407–447

Cambrian fans

  • Nathan Reading

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

    Massachusetts Institute of Technology, Cambridge, United States
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Abstract

For a finite Coxeter group~WW and a Coxeter element~cc of W,W, the cc-Cambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of~W ⁣W\!. Its maximal cones are naturally indexed by the cc-sortable elements of~W ⁣W\!. The main result of this paper is that the known bijection \clc\cl_c between cc-sortable elements and cc-clusters induces a combinatorial isomorphism of fans. In particular, the cc-Cambrian fan is combinatorially isomorphic to the normal fan of the generalized associahedron for~W ⁣W\!. The rays of the cc-Cambrian fan are generated by certain vectors in the WW-orbit of the fundamental weights, while the rays of the cc-cluster fan are generated by certain roots. For particular (``bipartite'') choices of~cc, we show that the cc-Cambrian fan is linearly isomorphic to the cc-cluster fan. We characterize, in terms of the combinatorics of clusters, the partial order induced, via the map \clc\cl_c, on cc-clusters by the cc-Cambrian lattice. We give a simple bijection from cc-clusters to cc-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\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