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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.

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)
Full Text: DOI arXiv Link
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