Wednesday, 18 April 2018
Thomas Gobet: Dual and classical generators of Artin groups of spherical type
The n-strand braid group has two Garside structures, a classical and a dual one, leading to different solutions to the word problem in the group. The corresponding Garside monoids are the classical braid monoid and the Birman-Ko-Lee braid monoid. These monoids exist for any Artin group of spherical type (i.e., attached to a finite Coxeter group).The aim of the talk is to explain how to express the generators of an Artin group of spherical type coming from the dual braid monoid (and more generally the simples, that is, the divisors of the Garside element of the dual monoid) in terms of the classical generators: we give a closed formula to express a simple of the dual braid monoid, using combinatorial objects called c-sortable elements (introduced by Reading). It gives a word of shortest possible length in the classical generators representing a simple element of the dual braid monoid. It has as an immediate consequence that the simples of the dual braid monoids are Mikado braids.