Dual and classical generators of Artin groups of spherical type

Please note the unusual time: just for this week, the algebra and topology seminar will be from 2:30 to 3:30pm.

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 properties generalize to Artin groups attached to finite Coxeter groups.

The aim of the talk is to explain how to express the generators of the braid group 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 monoid, using combinatorial objects called c-sortable elements (introduced by Reading). It gives a word of shortest possible length in the classical generators for any simple of the dual braid monoid. It has as an immediate consequence that the simples of the dual braid monoids are Mikado braids.