Cambrian fans

By Nathan Reading, David E. Speyer. 2006

96 citations recorded by the source corpus.

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 g-vectors and quasi-Cartan companions.

Subfields: Geometry and Topology, Mathematical Physics, Spectroscopy

View Paper: https://www.semanticscholar.org/paper/35e142bb823e72ce75f461b463cf95bec37cc0a7

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Mathematical Physics

In Spectroscopy

Citations in the corpus

Non-kissing complexes and tau-tilting for gentle algebras (2017, background): “in direction (1;1;1). Note that the g-vector fan is supported by the c-vector fan (right). For oriented paths, g-vector fans are known as type ACambrian fans, constructed by N. Reading and D. Speyer [RS09] in connection with type Acluster algebras [FZ02,FZ03]. Note that the topmost quiver of Figure4”
Cluster Algebras of Finite Type via Coxeter Elements and Principal Minors (2008, result): “ation-theoretic arguments in the above proof of (3.20). Remark 6.1. We have already mentioned in Remark 1.11 that the set Π(c) is identical with the set of generators of the c-Cambrian fan studied in [14]. In particular, our Theorem 1.10 implies [14, Theorem 10.2], proved there only modulo [10, Conj”
Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups (2006, background): “e corresponding Cluster complex. This conjecture has now been proved in types A and B by Hohlweg and Lange [69, Corollaries 4.1 and 4.2], and it has been proved in a uniform way by Reading and Speyer [90]. 5.2.3. The Generalized Cluster Complex. Finally, we consider a FussCatalan generalization of t”
Subword complexes, cluster complexes, and generalized multi-associahedra (2011, unknown): “In [50], N.”
Coxeter submodular functions and deformations of Coxeter permutahedra (2019, background): “Reading [37] introduced the Cambrian fan Fc, a complete fan with rich combinatorial properties and close connections with the theory of cluster algebras [38].”
Sortable elements in infinite Coxeter groups (2008, methodology): “. David E Speyer was supported by a Research Fellowship from the Clay Mathematics Institute. 1 2 NATHAN READING AND DAVID E SPEYER 1. Introduction This paper is the continuation of a series of papers [31, 32, 33, 34] devoted to the study of sortable elements and Cambrian lattices in a Coxeter group ”
Total positivity and cluster algebras (2010, background): “In particular, not covered here are the theory of cluster categories and the various facets of categorification [39, 40, 41, 50]; the connections between cluster algebras and Poisson geometry [32, 33]; closely related work on cluster varieties arising in higher Teichmüller theory [16, 17]; the polyh”
Brick polytopes of spherical subword complexes and generalized associahedra (2011, background): “n upper position in Iif r(I;i) 2 . Moreover, its corresponding almost positive root lrc(i) is an upper root in lrc(I). Compare the original equivalent denition of upper roots before Corollary 8.6 in [RS09]. Theorem 6.23. The bijection lp c= nc so is given by lpc : Facc(W) ! Ncpc(W) I 7! s r(I;j 1) s”
Universal geometric cluster algebras (2012, unknown): “The definitions and results given here are inspired by the juxtaposition of several results on cluster algebras of finite type from [12,19,23,30].”
Permutahedra and generalized associahedra (2007, background): “Acknowledgements The authors thank Nathan Reading and David Speyer for having made their results in [21] available to us during their work and for their helpful remarks.”
Acyclic Cluster Algebras Revisited (2012, result): “ation with the authors’ other work Both authors have written several other papers related to the present work. The notations in this paper are entirely compatible with the ﬁrs t author’s notations in [RS1] and [RS3]. In order to achieve this, it is necessary in Section 5 to deﬁne E([X],[Y])to be ∑(−”