The dual braid monoid

By David Bessis. 2001

363 citations recorded by the source corpus.

Abstract We study a new monoid structure for Artin groups associated with finite Coxeter systems. Like the classical positive braid monoid, the new monoid is a Garside monoid. We give several equivalent constructions: algebraically, the new monoid arises when studying Coxeter systems in a “dual” way, replacing the pair ( W , S ) by ( W , T ), with T the set of all reflections; geometrically, it arises when looking at the reflection arrangement from a certain basepoint. In the type A case, we recover the monoid constructed by Birman, Ko and Lee.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c16712899ad36bf75d370ebe51f1ce73631e94ea

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In Discrete Mathematics and Combinatorics

1. Nathan Reading
2. Drew Armstrong
3. David E. Speyer
4. I. Reiten
5. Bruce E. Sagan
6. Aslak Bakke Buan
7. Sergey Fomin
8. Ralf Schiffler

In Geometry and Topology

1. I. Reiten
2. B. Keller
3. Ralf Schiffler
4. Nathan Reading
5. Aslak Bakke Buan
6. David E. Speyer
7. Osamu Iyama
8. Drew Armstrong

Citations in the corpus

• CLUSTER MUTATION VIA QUIVER REPRESENTATIONS (2004, background): “rators of a cluster algebra play an important role. Cluster algebras have also been related to Poisson geometry [GSV1], Teichmu¨ller spaces [GSV2], positive spaces and stacks [FG], dual braid monoids [BES], ad-nilpotent ideals of a Borel subalgebra of a simple Lie algebra [P] as well as representati”
• K(π, 1) for Artin Groups of Finite Type (2002, background): “If σ′ is balanced define the Cn element σ by σ = [1]σ′.”
• Noncrossing partitions and the shard intersection order (2009, neutral): “Work of Athanasiadis, Bessis, Biane, Brady, Reiner and Watt [4, 5, 6, 9, 40] led to the recognition that the classical noncrossing partitions are a special case ( W = S n ) of a combinatorial construction which yields a noncrossing partition lattice NC( W ) for each finite Coxeter group W .”
• Cluster algebras: Notes for the CDM-03 conference (2003, background): “These numbers are known to count [6, 12, 14, 29, 34, 35, 45] a variety of combinatorial objects related to the root system Φ.”
• Braids: A Survey (2004, background): “We also thank Robert Bell, David Bessis, Nathan Broaddus, John Cannon, Ruth Charney, Fred Cohen, Patrick Dehornoy, Roger Fenn, Daan Krammer, Lee Mosher, Luis Paris, Richard Stanley, Morwen Thistlethwaite and Bert Wiest for their help in chasing down facts and references, and helping us fill gaps in ”
• Surveys in Combinatorics 2011: The cyclic sieving phenomenon: a survey (2010, background): “As defined by Brady and Watt in the real case [10] and Bessis in the complex [6, 7], the noncrossing elements in W are NC(W ) = {w ∈W : lR(w) + lR(w g) = lR(g)}.”
• A categorification of non-crossing partitions (2013, methodology): “This theorem is an analogue (for Weyl groups of symmetrisable Kac–Moody Lie algebras) of a result of Bessis [4] which describes for finite Coxeter groups the non-crossing partitions as Coxeter elements of parabolic subgroups.”
• Artin groups of Euclidean type (2013, methodology): “In particular, this is known to hold for all spherical Artin groups [3,9] and we prove it here for all Euclidean Artin groups as our third main result.”
• Parking Spaces (2012, methodology): “We hope that some of our conjectures can be extended to complex reflection groups, at least in the well-generated case considered by Bessis and Reiner [11], and perhaps even in the arbitrary case considered by Gordon and Griffeth [28].”
• Finite complex reflection arrangements are $K(\pi,1)$ (2006, neutral): “When W is complexified real, a dual braid monoid was constructed in [3] (generalising the construction of Birman-Ko-Lee [11]; similar partial results were independently obtained by Brady and Watt [14]).”
• Conjugacy in Garside groups III: Periodic braids (2006, neutral): “Nevertheless, we will not use the classical Garside structure on the Artin–Tits group A(Bn−1), but the dual Garside structure defined in [2].”
• Generalized Catalan Numbers, Weyl Groups and Arrangements of Hyperplanes (2004, background): “It is striking that, apart from regions of CatΦ, antichains in the root poset of Φ, and W -orbits of Q̌/(h + 1)Q̌, the product (4) is also known to count noncrossing partitions associated to W (see [5, 18]) and the clusters associated to Φ by Fomin and Zelevinsky [10] or, equivalently, vertices of t”
• K(π 1)'s for Artin Groups of Finite Type (2000, background): “If σ′ is balanced define the Cn element σ by σ = [1]σ′.”