Springer theory in braid groups and the Birman-Ko-Lee monoid

By David Bessis, F. Digne, Jean Michel. 2000

103 citations recorded by the source corpus.

We state a conjecture about centralizers of certain roots of central elements in braid groups, and check it for braid groups of type A, B, G(d, 1, r) and a couple of other cases. Our proof makes use of results from Birman-Ko-Lee, of which we give a new intrinsic account.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c066eb187425e63348c58af575fbc2b2c59e8f08

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups (2006, background): “Bessis, Digne and Michel were able to switch the terminology from “nonobstructing” to “non-crossing” in their paper [22] before publication.”
Finite complex reection arrangements are K(; 1) (2015, unknown): “1”
GEOMETRIC PRESENTATIONS FOR THE PURE BRAID GROUP (2006, background): “The n-string braid group Braidn can be viewed as the fundamental group of the configuration space of n distinct but indistinguishable points in a disc: Braidn ∼= π1(C(D, n)).”
Complete positive group presentations (2001, background): “nd complemented, and the cube condition is satisﬁed, as established in [3]. So the presentation is complete, as are more generally the so-called dual presentations of the Artin groups investigated in [2, 32]. Let us mention that other criteria have been established subsequently, always in the comple”
Conjugacy in Garside groups I: cyclings, powers and rigidity (2006, background): “If it is periodic, then its centralizer is well known [7] and easy to compute, with the help of the algorithm in [10].”
The language of geodesics for Garside groups (2004, unknown): “(This is directly related to the Birman-Ko-Lee monoid for braid groups [2], [3].”
The K(π,1)-conjecture for the affine braid groups (2003, background): “The element ¢ projects to the longest element of the Coxeter group W corresponding to A. More recently, a difierent Garside structure for these groups was introduced by Bessis, Digne, and Michel in [ BDM ] and [Be].”
The dual braid monoid (2001, methodology): “In [ 2 ] is mentioned a conjecture about centralizers in generalized braid groups of certain dth roots of central elements, and the Birman–Ko–Lee monoid is used to prove the conjecture for the type A case.”
Non-crossing partition lattices in finite real reflection groups (2007, unknown): “The second Garside structure for B n is described in [2] and the structure for general ﬁnite W in [3].”
Finite complex reflection arrangements are $K(\pi,1)$ (2006, unknown): “Combined with [7] and [8], this completes the longstanding task of finding presentations for all generalized braid groups associated with finite complex reflection groups.”
Conjugacy in Garside groups III: Periodic braids (2006, unknown): “Fortunately, it is shown in [3] that two symmetric braids are isotopic in B 2n−2 if and only if they are isotopic through symmetric braids.”
Bestvina's Normal Form Complex and the Homology of Garside Groups (2002, background): “ds whose groups of fractions are the Artin groups of ﬁnite type. The most well-known of these is the Birman-Ko-Lee braid monoid [5] which was shown by Bessis, Digne, and Michel to be a Garside monoid [2]. This has been generalized by Bessis, Brady, and Brady and Watt, to all Artin groups of ﬁnite ty”
Derived equivalences and finite dimensional algebras (2006, unknown): “There is a morphism Bd → CB(wd), uniquely defined up to conjugation by an element of the pure braid group of CW(wd) (it is expected to be an isomorphism, and known to be such in a number of cases [8]).”
Basic results on braid groups (2010, background): “The centralizer of these braids, determined in [7] are precisely the braids which are invariant under that rotation.”