Simple dual braids, noncrossing partitions and Mikado braids of type Dn

Baumeister, Barbara; Gobet, Thomas

doi:10.1112/blms.12100

Cited by 6 publications

(11 citation statements)

References 16 publications

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“…This result was conjectured in a joint work with Digne [19], where it was proven for every irreducible W except in type D n . Type D n was proven in a joint work with Baumeister [4] and independently, Licata and Queffelec [34] proved it in types A n , D n and E n . The proofs there use either topological realizations of the braid groups (i.e., in terms of Artin braids) or categorification techniques.…”

Section: Introductionmentioning

confidence: 93%

Dual Garside structures and Coxeter sortable elements

Gobet¹

2018

Preprint

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In Artin-Tits groups attached to Coxeter groups of spherical type, we give a combinatorial formula to express the simple elements of the dual braid monoids in the classical Artin generators. Every simple dual braid is obtained by lifting an S-reduced expression of its image in the Coxeter group, in a way which involves Reading's c-sortable elements. It has as an immediate consequence that simple dual braids are Mikado braids (the known proofs of this result either require topological realizations of the Artin groups or categorification techniques), and hence that their images in the Iwahori-Hecke algebras have positivity properties. In the classical types, this requires to give an explicit description of the inverse of Reading's bijection from c-sortable elements to noncrossing partitions of a Coxeter element c, which might be of independent interest. The bijections are described in terms of the noncrossing partition models in these types. While the proof of the formula is case-by-case, it is entirely combinatorial and we develop an approach which reduces a uniform proof to uniformly proving a lemma about inversion sets of c-sortable elements.

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Section: Introductionmentioning

confidence: 93%

Dual Garside structures and Coxeter sortable elements

Gobet¹

2018

Preprint

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“…Problem As the graph of irreducible parabolic subgroups of a dihedral Artin–Tits group is not connected, the only infinite family of Artin–Tits groups of spherical type for which the hyperbolicity of the graph of irreducible parabolic subgroups is still open is the type

D_{n}

. The Artin–Tits group of type

D_{n}

can be seen as an index 2 subgroup of the quotient of an Artin–Tits group of type

B_{n}

by the normal subgroup generated by the standard generator

τ_{1}

[2]. It is interesting to ask whether this embedding can be used to establish the hyperbolicity of

{scriptC}_{p a r a b} false(D_{n} false)

.…”

Section: Hyperbolic Structures On Artin–tits Groupsmentioning

confidence: 99%

Curve graphs for Artin–Tits groups of type B, A∼ and C∼ are hyperbolic

Calvez

Cruz

2021

Trans London Maths Soc

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The graph of irreducible parabolic subgroups is a combinatorial object associated to an Artin-Tits group A defined so as to coincide with the curve graph of the (n + 1)-times punctured disk when A is Artin's braid group on (n + 1) strands. In this case, it is a hyperbolic graph, by the celebrated Masur-Minsky's theorem. Hyperbolicity of the graph of irreducible parabolic subgroups for more general Artin-Tits groups is an important open question. In this paper, we give a partial affirmative answer.For n 3, we show that the graph of irreducible parabolic subgroups associated to the Artin-Tits group of spherical type Bn is also isomorphic to the curve graph of the (n + 1)-times punctured disk; hence, it is hyperbolic.For n 2, we show that the graphs of irreducible parabolic subgroups associated to the Artin-Tits groups of euclidean type An and Cn are isomorphic to some subgraphs of the curve graph of the (n + 2)-times punctured disk which are not quasi-isometrically embedded. We prove nonetheless that these graphs are hyperbolic.

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“…Note that by (9), Let s be a semi-circular element as in (9). Then the moment generating functions of s and s 2 are given by…”

Section: Non-crossing Partitions In Free Probabilitymentioning

confidence: 99%

“…that the rescaled sum (a 1 + a 2 )/ √ 2 of two free elements a 1 , a 2 of a non-commutative probability space (A, ϕ) which both have density w(x) again has a Wigner distribution. In free probability an element s of (A, ϕ) with density w(x) is called semi-circular and its moments are given by (9) ϕ(s n ) =…”

Section: Non-crossing Partitions In Free Probabilitymentioning

confidence: 99%

“…There is an embedding of Artin groups of type B n into those of type A 2n−1 . The situation in type D n is as follows [9]: The root system of type D n embeds into the root system of type B n , which implies that the Coxeter system of type D n is a subsystem of that one of type B n . But there is not an embedding of the Artin group of type D n into that one of type B n that satisfies a certain natural condition.…”

Section: Non-crossing Partitions In Coxeter Groupsmentioning

confidence: 99%

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Non-crossing partitions

Baumeister¹,

Bux²,

Götze³

et al. 2019

EMS Series of Congress Reports

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Non-crossing partitions have been a staple in combinatorics for quite some time. More recently, they have surfaced (sometimes unexpectedly) in various other contexts from free probability to classifying spaces of braid groups. Also, analogues of the non-crossing partition lattice have been introduced. Here, the classical non-crossing partitions are associated to Coxeter and Artin groups of type An, which explains the tight connection to the symmetric groups and braid groups. We shall outline those developments.

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Simple dual braids, noncrossing partitions and Mikado braids of type Dn

Cited by 6 publications

References 16 publications

Dual Garside structures and Coxeter sortable elements

Dual Garside structures and Coxeter sortable elements

Curve graphs for Artin–Tits groups of type B, A∼ and C∼ are hyperbolic

Non-crossing partitions

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