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Cambrian fans. (English) Zbl 1213.20038
Summary: 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 \(\text{cl}_c\) 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 \(\text{cl}_c\), 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.

MSC:
20F55 Reflection and Coxeter groups (group-theoretic aspects)
13F60 Cluster algebras
05E15 Combinatorial aspects of groups and algebras (MSC2010)
06A07 Combinatorics of partially ordered sets
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
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