## Non-crossing partition lattices in finite real reflection groups

Citations: | 12 - 2 self |

### BibTeX

@MISC{Brady_non-crossingpartition,

author = {Thomas Brady and Colum Watt},

title = {Non-crossing partition lattices in finite real reflection groups},

year = {}

}

### OpenURL

### Abstract

Abstract. For a finite real reflection group W with Coxeter element γ we give a case-free proof that the closed interval, [I,γ], forms a lattice in the partial order on W induced by reflection length. Key to this is the construction of an isomorphic lattice of spherical simplicial complexes. We also prove that the greatest element in this latter lattice embeds in the type W simplicial generalised associahedron, and we use this fact to give a new proof that the geometric realisation of this associahedron is a sphere. 1.

### Citations

148 |
Sur les partitions non croisées d’un cycle
- Kreweras
- 1972
(Show Context)
Citation Context ... in [8], and the Cn and Dn groups in [10]. Bessis treats all cases in [3]. Since the lattice in the symmetric group case coincides with the lattice of noncrossing partitions introduced by Kreweras in =-=[16]-=-, it has become customary to call Received by the editors January 27, 2005 and, in revised form, December 17, 2005. 2000 Mathematics Subject Classification. Primary 20F55; Secondary 05E15. 1983 c○2007... |

125 | The braid group and other groups, Quart - Garside - 1969 |

123 | A new approach to the word and conjugacy problems in the braid groups
- Birman, Ko, et al.
- 1998
(Show Context)
Citation Context ..., but Daan Krammer has used it independently in unpublished work. In the case of the braid group Bn, whereWis the symmetric group Σn, the larger generating set coincides with the band generators from =-=[5]-=-. The second Garside structure for Bn is described in [2] and the structure for general finite W in [3]. The general construction of K(A(W ), 1)’s is described in [11], using ideas from [4]. A central... |

110 | Non-crossing partitions for classical reflection groups, preprint dated
- Reiner
- 1996
(Show Context)
Citation Context ...5. 1983 c○2007 American Mathematical Societys1984 T. BRADY AND C. WATT this lattice, in the case of general W ,thetypeW non-crossing partition lattice. The type Cn and type Dn lattices are studied in =-=[17]-=- and [1], respectively. In this paper we give a new proof of the lattice property that is independent of the classification of finite real reflection groups. For this purpose we introduce a new simpli... |

106 |
Lie groups and Lie algebras. Chapters 4–6
- Bourbaki
- 2002
(Show Context)
Citation Context ...uction on i, startingati = t. �s1994 T. BRADY AND C. WATT Note 5.6. The proofs of Lemma 5.5 and Theorem 5.4 can be shortened considerably by using the notion of a reflection ordering as introduced in =-=[6]-=- and developed in [12]. It follows from the definition of ρi together with Exercise 6.2 of Chapter 5 of [6] and Proposition 2.13 of [12] that R(ρ1),R(ρ2),...,R(ρ nh/2) is a reflection ordering on W , ... |

48 |
Reflection groups and invariant theory
- Kane
- 2001
(Show Context)
Citation Context ...F (α). This equation follows from Corollary 1 of [9]. If R(�v) denotes the reflection in �v ⊥ ,then (3.2) �v ∈ M(α) ⇒ R(�v)α = αR[α −1 (�v)]. This equation follows from Property (A-4) of Chapter 1 of =-=[15]-=-. (3.3) If α ≤ β ≤ δ, then α −1 β ≤ α −1 δ and βα −1 ≤ δα −1 . The first part of this equation follows from Proposition 3 of [9] while the second part follows from the fact that conjugation in W prese... |

47 |
The dual braid monoid, Ann
- Bessis
- 2003
(Show Context)
Citation Context ...ereWis the symmetric group Σn, the larger generating set coincides with the band generators from [5]. The second Garside structure for Bn is described in [2] and the structure for general finite W in =-=[3]-=-. The general construction of K(A(W ), 1)’s is described in [11], using ideas from [4]. A central result needed in the development of this parallel theory is that the closed interval, [I,γ], bounded b... |

41 | Springer theory in braid groups and the BirmanKo-Lee monoid
- Bessis, Digne, et al.
(Show Context)
Citation Context ...ed work. In the case of the braid group Bn, whereWis the symmetric group Σn, the larger generating set coincides with the band generators from [5]. The second Garside structure for Bn is described in =-=[2]-=- and the structure for general finite W in [3]. The general construction of K(A(W ), 1)’s is described in [11], using ideas from [4]. A central result needed in the development of this parallel theory... |

38 | Noncrossing partitions for the group Dn
- Athanasiadis, Reiner
(Show Context)
Citation Context ...○2007 American Mathematical Societys1984 T. BRADY AND C. WATT this lattice, in the case of general W ,thetypeW non-crossing partition lattice. The type Cn and type Dn lattices are studied in [17] and =-=[1]-=-, respectively. In this paper we give a new proof of the lattice property that is independent of the classification of finite real reflection groups. For this purpose we introduce a new simplicial com... |

38 | Bestvina’s normal form complex and the homology of Garside groups - Charney, Meier, et al. - 2002 |

36 |
Hecke algebras and shellings of Bruhat intervals
- Dyer
- 1993
(Show Context)
Citation Context ...ati = t. �s1994 T. BRADY AND C. WATT Note 5.6. The proofs of Lemma 5.5 and Theorem 5.4 can be shortened considerably by using the notion of a reflection ordering as introduced in [6] and developed in =-=[12]-=-. It follows from the definition of ρi together with Exercise 6.2 of Chapter 5 of [6] and Proposition 2.13 of [12] that R(ρ1),R(ρ2),...,R(ρ nh/2) is a reflection ordering on W , in the sense that the ... |

35 | K(π,1)’s for Artin groups of finite type
- Brady, Watt
(Show Context)
Citation Context ...the lattice property use the classification of finite real reflection groups with different methods applied to the different groups. The symmetric group is handled in [8], and the Cn and Dn groups in =-=[10]-=-. Bessis treats all cases in [3]. Since the lattice in the symmetric group case coincides with the lattice of noncrossing partitions introduced by Kreweras in [16], it has become customary to call Rec... |

33 | Non-positively curved aspects of Artin groups of finite type, Geometry & Topology 3
- Bestvina
- 1999
(Show Context)
Citation Context ...ators from [5]. The second Garside structure for Bn is described in [2] and the structure for general finite W in [3]. The general construction of K(A(W ), 1)’s is described in [11], using ideas from =-=[4]-=-. A central result needed in the development of this parallel theory is that the closed interval, [I,γ], bounded by the identity I and a Coxeter element γ, formsa lattice in the partial order on W ind... |

29 |
A partial order on the symmetric group and new K(π,1)’s for the braid groups, to appear
- Brady
(Show Context)
Citation Context ...ew K(π, 1)’s. Existing proofs of the lattice property use the classification of finite real reflection groups with different methods applied to the different groups. The symmetric group is handled in =-=[8]-=-, and the Cn and Dn groups in [10]. Bessis treats all cases in [3]. Since the lattice in the symmetric group case coincides with the lattice of noncrossing partitions introduced by Kreweras in [16], i... |

29 |
Finite reflection groups
- Steinberg
- 1959
(Show Context)
Citation Context ... the set of roots. We define the simplicial complex X(γ) and give an example. We also collect results about the partial order on the orthogonal group. In Section 4 we extend some of the material from =-=[18]-=- and analyse the dot products of roots of W with elements in the γ orbits of vectors in the dual basis. In Section 5 we find simple systems for certain subgroups and describe X(σ)whenσ has length two.... |

12 |
Artin groups of finite type with three generators
- Brady
(Show Context)
Citation Context ...ositive monoid embeds in A(W ) and new K(π, 1)’s for A(W ) have been constructed. The larger generating set gives A(W ) a second structure as a Garside group. The larger generating set is proposed in =-=[7]-=-, but Daan Krammer has used it independently in unpublished work. In the case of the braid group Bn, whereWis the symmetric group Σn, the larger generating set coincides with the band generators from ... |

1 |
Y-systems and generalised
- Fomin, Zelevinsky
(Show Context)
Citation Context ...ise the geometric realisation of X(σ)and we prove the lattice property in Section 7. We conclude in Section 8 by explaining the connection between our construction and the generalised associahedra of =-=[13]-=-. In the process, we give a new proof that the latter is a spherical simplicial complex. 2. Motivation The idea behind the proof of the lattice property can be described without many of the technicali... |