Realizations of the Associahedron and Cyclohedron

Hohlweg, Christophe; Langé, Christine

doi:10.1007/s00454-007-1319-6

Cited by 91 publications

(190 citation statements)

References 27 publications

(51 reference statements)

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“…This last result has been established by Hohlweg and Lange [17] for types A and B and will be proven for all types in a future paper by Hohlweg, Lange and Thomas [18].…”

Section: Introductionsupporting

confidence: 65%

Cambrian fans

Reading¹,

Speyer²

2009

J. Eur. Math. Soc.

156

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Abstract. 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 Worbit 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.

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“…This last result has been established by Hohlweg and Lange [17] for types A and B and will be proven for all types in a future paper by Hohlweg, Lange and Thomas [18].…”

Section: Introductionsupporting

confidence: 65%

Cambrian fans

Reading¹,

Speyer²

2009

J. Eur. Math. Soc.

156

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show abstract

“…In a subsequent paper, C. Stump and V. Pilaud obtain a geometric construction of a class of subword complexes containing generalized associahedra purely in terms of subword complexes [45]. Clusters [20,[47][48][49] Generalized associahedron [12,26,45,61] Multi-clusters (present paper) Generalized multi-associahedron (existence conjectured)…”

Section: Generalized Multi-associahedra and Polytopality Of Sphericalmentioning

confidence: 99%

“…Its facets correspond to triangulations (i.e., maximal subsets of diagonals which are mutually noncrossing). This simplicial complex is the boundary complex of the dual associahedron [13,22,25,26,37,38], we refer to the recent book [39] for a detailed treatment of the history of associahedra. The complex Δ m can be generalized using a positive integer k with 2k + 1 ≤ m: define a (k + 1)-crossing to be a set of k + 1 diagonals which are pairwise crossing.…”

Section: Multi-triangulationsmentioning

confidence: 99%

Subword complexes, cluster complexes, and generalized multi-associahedra

2013

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In this paper, we use subword complexes to provide a uniform approach to finite-type cluster complexes and multi-associahedra. We introduce, for any finite Coxeter group and any nonnegative integer k, a spherical subword complex called multi-cluster complex. For k = 1, we show that this subword complex is isomorphic to the cluster complex of the given type. We show that multi-cluster complexes of types A and B coincide with known simplicial complexes, namely with the simplicial complexes of multi-triangulations and centrally symmetric multi-triangulations, respectively. Furthermore, we show that the multi-cluster complex is universal in the sense that every spherical subword complex can be realized as a link of a face of the multi-cluster complex.

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“…A type B associahedron was defined by Simion [95] (see also [67]). For many beautiful pictures of associahedra, see the survey [52] by Fomin and Reading.…”

Section: Polygon Triangulations and The Associahedronmentioning

confidence: 99%

Generalized noncrossing partitions and combinatorics of Coxeter groups

2009

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under whose guidance the original draft [2] was written. I thank my wife Heather for her thorough reading of the manuscript and for helpful mathematical discussions. Thanks to the anonymous referee for many valuable suggestions. And thanks to Hugh Thomas who made several contributions to my understanding; in particular, he suggested the proof of Theorem 3.7.2.I have benefitted from valuable conversations with many mathematicians; their suggestions have improved this memoir in countless ways. Thanks to: (in alphabetical order)

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Realizations of the Associahedron and Cyclohedron

Cited by 91 publications

References 27 publications

Cambrian fans

Cambrian fans

Subword complexes, cluster complexes, and generalized multi-associahedra

Generalized noncrossing partitions and combinatorics of Coxeter groups

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