Noncrossing Partitions in Surprising Locations

McCammond, Jon

doi:10.2307/27642003

Cited by 50 publications

(44 citation statements)

References 27 publications

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“…For an account of the history of noncrossing partitions, see the extensive survey by Simion [96]. A more modern perspective is given by McCammond [77].…”

Section: Symmetries Of Regular Polytopesmentioning

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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“…For an account of the history of noncrossing partitions, see the extensive survey by Simion [96]. A more modern perspective is given by McCammond [77].…”

Section: Symmetries Of Regular Polytopesmentioning

confidence: 99%

Generalized noncrossing partitions and combinatorics of Coxeter groups

2009

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“…Each of the five possible posets (corresponding to the finite Coxeter groups of type A 4 , B 4 , D 4 , F 4 and H 4 ) is known to be a lattice and thus by Proposition 7.4 we only need to check whether or not NC W contain a global spindle of girth 6 whose elements alternate between adjacent ranks. Moreover, because NC W is self-dual [17], it is sufficient to search for the configuration on the lefthand side of Figure 9. The second author wrote a short program in GAP to construct these posets and to search for this particular configuration.…”

Section: Artin Groupsmentioning

confidence: 99%

Braids, posets and orthoschemes

Brady

McCammond

2010

Algebr. Geom. Topol.

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In this article we study the curvature properties of the order complex of a bounded graded poset under a metric that we call the "orthoscheme metric". In addition to other results, we characterize which rank 4 posets have CAT.0/ orthoscheme complexes and by applying this theorem to standard posets and complexes associated with four-generator Artin groups, we are able to show that the 5-string braid group is the fundamental group of a compact nonpositively curved space.05E15, 06A06, 20F36, 20F65, 51M20; 06A11Barycentric subdivision subdivides an n-cube into isometric metric simplices called orthoschemes. We use orthoschemes to turn the order complex of a graded poset P into a piecewise Euclidean complex K that we call its orthoscheme complex. Our goal is to investigate the way that combinatorial properties of P interact with curvature properties of K . More specifically, we focus on combinatorial configurations in P that we call spindles and conjecture that they are the only obstructions to K being CAT.0/.Poset Curvature Conjecture The orthoscheme complex of a bounded graded poset P is CAT.0/ if and only if P has no short spindles.One way to view this conjecture is as an attempt to find something like the flag condition that tests whether a cube complex is CAT.0/. We highlight this perspective in Section 7. Our main theorem establishes the conjecture for posets of low rank.Theorem A The orthoscheme complex of a bounded graded poset P of rank at most 4 is CAT.0/ if and only if P has no short spindles.Using Theorem A, we prove that the 5-string braid group, also known as the Artin group of type A 4 , is a CAT.0/ group. More precisely, we prove the following.Theorem B Let K be the Eilenberg-Mac Lane space for a four-generator Artin group of finite type built from the corresponding poset of W-noncrossing partitions and endowed with the orthoscheme metric. When the group is of type A 4 or B 4 , the complex K is CAT.0/ and the group is a CAT.0/ group. When the group is of type D 4 , F 4 or H 4 , the complex K is not CAT.0/.

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“…Specifically, noncrossing partitions are a powerful tool in the theory of Artin groups [4,7]. For an accessible introduction to this application, focusing on the special case of the symmetric group (and thus the braid group), see [23], which also discusses other applications of noncrossing partitions to free probability and combinatorics. The definitions of both noncrossing partitions and clusters involve the choice of a Coxeter element c for W, and to emphasize this fact we will refer to them as c-noncrossing partitions and c-clusters.…”

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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Noncrossing Partitions in Surprising Locations

Cited by 50 publications

References 27 publications

Generalized noncrossing partitions and combinatorics of Coxeter groups

Generalized noncrossing partitions and combinatorics of Coxeter groups

Braids, posets and orthoschemes

Cambrian fans

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