Simple dual braids, noncrossing partitions and Mikado braids of type $D_n$

Barbara Baumeister; Thomas Gobet

doi:10.1112/blms.12100

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arXiv · 1703.09599

Simple dual braids, noncrossing partitions and Mikado braids of type $D_n$

Barbara Baumeister·Thomas Gobet

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Abstract

We show that the simple elements of the dual Garside structure of an Artin group of type $D_n$ are Mikado braids, giving a positive answer to a conjecture of Digne and the second author. To this end, we use an embedding of the Artin group of type $D_n$ in a suitable quotient of an Artin group of type $B_n$ noticed by Allcock, of which we give a simple algebraic proof here. This allows one to give a characterization of the Mikado braids of type $D_n$ in terms of those of type $B_n$ and also to describe them topologically. Using this topological representation and Athanasiadis and Reiner's model for noncrossing partitions of type $D_n$ which can be used to represent the simple elements, we deduce the above mentioned conjecture.

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BibTeX RIS

Barbara Baumeister, Thomas Gobet. 2017-03-28. Simple dual braids, noncrossing partitions and Mikado braids of type $D_n$. https://doi.org/10.1112/blms.12100

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