Non-crossing partition lattices in finite real reflection groups

Brady, Thomas A.; Watt, Colum

doi:10.1090/s0002-9947-07-04282-1

Cited by 82 publications

(197 citation statements)

References 20 publications

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“…These theorems result from our formulae for the (combinatorial) decomposition numbers upon appropriate summations. Subsequently, it is shown that the corresponding formulae imply all known enumeration results on non-crossing partitions and generalised non-crossing partitions, plus several new ones; see Corollaries 12,14,[16][17][18][19] and the accompanying remarks. Section 9 presents the announced computational proof of the F = M (ex-)Conjecture for type D n , based on our formula in Corollary 19 for the rank-selected chain enumeration in the poset of generalised non-crossing partitions of type D n , while Section 10 addresses Conjecture 3.5.13 from [1], showing that it does not hold in general since it fails in type D n .…”

Section: Introductionmentioning

confidence: 93%

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Decomposition numbers for finite Coxeter groups and generalised non-crossing partitions

Krattenthaler¹,

Müller

2009

Trans. Amer. Math. Soc.

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These results are then used to determine, for a fixed positive integer l and fixed integers r 1 ≤ r 2 ≤ · · · ≤ r l , the number of multi-chains π 1 ≤ π 2 ≤ · · · ≤ π l in Armstrong's generalised non-crossing partitions poset, where the poset rank of π i equals r i and where the "block structure" of π 1 is prescribed. We demonstrate that this result implies all known enumerative results on ordinary and generalised non-crossing partitions via appropriate summations. Surprisingly, this result on multi-chain enumeration is new even for the original non-crossing partitions of Kreweras. Moreover, the result allows one to solve the problem of rank-selected chain enumeration in the type D n generalised non-crossing partitions poset, which, in turn, leads to a proof of Armstrong's F = M Conjecture in type D n , thus completing a computational proof of the F = M Conjecture for all types. It also allows one to address another conjecture of Armstrong on maximal intervals containing a random multi-chain in the generalised non-crossing partitions poset.

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Section: Introductionmentioning

confidence: 93%

“…[2,3,4,5,8,9,14,15,16,17,20]). They reduce to the classical non-crossing partitions of Kreweras [30] for the irreducible reflection groups of type A n (i.e., the symmetric groups) and to Reiner's [32] type B n non-crossing partitions for the irreducible reflections groups of type B n .…”

Section: Introductionmentioning

confidence: 99%

Decomposition numbers for finite Coxeter groups and generalised non-crossing partitions

Krattenthaler¹,

Müller

2009

Trans. Amer. Math. Soc.

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“…Another connection between noncrossing partitions and clusters has arisen recently. 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.…”

Section: Introductionmentioning

confidence: 99%

“…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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“…Section 2 includes background on simplicial complexes, (generalized) cluster complexes and a related partial order on a finite real reflection group. In particular, a new characterization (Theorem 2.3) of ∆ m (Φ), due to the second author [20,21], is reviewed, generalizing the one for ∆(Φ) given by T. Brady and C. Watt [9,Section 8]. The proof of Theorem 1.1, which relies on this characterization, is given in Section 3.…”

Section: Introductionmentioning

confidence: 99%

Shellability and higher Cohen-Macaulay connectivity of generalized cluster complexes

Athanasiadis¹,

Tzanaki

2008

Isr. J. Math.

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Abstract. Let Φ be a finite root system of rank n and let m be a nonnegative integer. The generalized cluster complex ∆ m (Φ) was introduced by S. Fomin and N. Reading. It was conjectured by these authors that ∆ m (Φ) is shellable and by V. Reiner that it is (m + 1)-Cohen-Macaulay, in the sense of Baclawski. These statements are proved in this paper. Analogous statements are shown to hold for the positive part ∆ m + (Φ) of ∆ m (Φ). An explicit homotopy equivalence is given between ∆ m + (Φ) and the poset of generalized noncrossing partitions, associated to the pair (Φ, m) by D. Armstrong.

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Non-crossing partition lattices in finite real reflection groups

Cited by 82 publications

References 20 publications

Decomposition numbers for finite Coxeter groups and generalised non-crossing partitions

Decomposition numbers for finite Coxeter groups and generalised non-crossing partitions

Cambrian fans

Shellability and higher Cohen-Macaulay connectivity of generalized cluster complexes

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