Shellability of noncrossing partition lattices

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

doi:10.1090/s0002-9939-06-08534-0

Cited by 51 publications

(117 citation statements)

References 19 publications

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“…, n, and drawing an oriented edge from vertex i to vertex j = i whenever π(i) = j. For example, Figure 1.2 displays the cycle diagram of the permutation (124)(376)(58) of the set [8]. Notice that we can associate each cycle of π with the convex hull of its vertices.…”

Section: Much Of the Theory Of These Posets Was Developed Independentmentioning

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: Much Of the Theory Of These Posets Was Developed Independentmentioning

confidence: 99%

Generalized noncrossing partitions and combinatorics of Coxeter groups

2009

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“…[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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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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“…W −0 is homotopy equivalent to a wedge of (n − 1)-dimensional spheres. With some more work, and using the previous statement as well as the main result of [4], one can show that the poset L n , to the number of facets of ∆ + (Φ) (positive clusters associated to Φ).…”

Section: Generalized Noncrossing Partitionsmentioning

confidence: 92%

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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Shellability of noncrossing partition lattices

Abstract: Abstract. We give a case-free proof that the lattice of noncrossing partitions associated to any finite real reflection group is EL-shellable. Shellability of these lattices was open for the groups of type Dn and those of exceptional type and rank at least three.

Cited by 51 publications

References 19 publications

Generalized noncrossing partitions and combinatorics of Coxeter groups

Generalized noncrossing partitions and combinatorics of Coxeter groups

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

Shellability and higher Cohen-Macaulay connectivity of generalized cluster complexes

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