h-vectors of generalized associahedra and noncrossing partitions

Watt, Colum; Brady, Thomas A.; Athanasiadis, Christos A.; McCammond, Jon

doi:10.1155/imrn/2006/69705

Cited by 23 publications

(55 citation statements)

References 13 publications

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“…In particular, Chapoton provides several heuristic arguments for the conjecture in [39]. Also in the case k = 1, Athanasiadis recently proved Chapoton's conjecture for M and F [6], and this proof is uniform when combined with recent results in [7]. The case of the H-triangle is still not well understood.…”

Section: Chapoton Trianglesmentioning

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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Section: Chapoton Trianglesmentioning

confidence: 99%

Generalized noncrossing partitions and combinatorics of Coxeter groups

2009

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“…Note that subwords come with their embedding into Q; two subwords P and P representing the same word are considered to be different if they involve generators at different positions within Q. In Example 2.1, we have seen an instance of a subword complex with Q = (q 1 Subword complexes are known to be vertex-decomposable and hence shellable [34,Theorem 2.5]. Moreover, they are topologically spheres or balls depending on the Demazure product of Q.…”

Section: Subword Complexesmentioning

confidence: 99%

“…We remark that this is not possible in general: in type B 3 with k = 2, as in Example 2.12, Δ 2 c (B 3 ) is isomorphic to the simplicial complex of centrally symmetric 2-triangulations of a regular convex 10-gon. Every pair of elements in the set A = { [1,4] sym , [4,7] sym , [7,10] sym } is contained in a minimal nonface. But since A does not contain a 3-crossing, it forms a face of Δ 2 c (B 3 ).…”

Section: Open Problem 92 Find Multi-catalan Polynomialsmentioning

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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“…Brady and Watt [9] construct a simplicial fan associated to c-noncrossing partitions (for bipartite c) and extend their construction to produce the c-cluster fan. Athanasiadis, Brady, McCammond and Watt [1] use the construction of [9] to give a bijection between clusters and noncrossing partitions. Their proof uses no type by type arguments and provides a different bijective proof that the k th entry of the h-vector of the c-cluster fan coincides with the number of c-noncrossing partitions of rank k. The bijection of [1] incorporates elements which are similar in appearance to the constructions of the present paper (see Remark 11.5), but many details of the relation between the two theories remain unclear.…”

Section: Introductionmentioning

confidence: 99%

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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h-vectors of generalized associahedra and noncrossing partitions

Cited by 23 publications

References 13 publications

Generalized noncrossing partitions and combinatorics of Coxeter groups

Generalized noncrossing partitions and combinatorics of Coxeter groups

Subword complexes, cluster complexes, and generalized multi-associahedra

Cambrian fans

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