Cambrian fans

• Autores: Nathan Reading, David E. Speyer
• Localización: Journal of the European Mathematical Society, ISSN 1435-9855, Vol. 11, Nº 2, 2009, págs. 407-447
• Idioma: inglés
• DOI: 10.4171/jems/155
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• Resumen

  • 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.