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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Interval Garside structures for the complex braid groups $ B(e,e,n)$


Author: Georges Neaime
Journal: Trans. Amer. Math. Soc. 372 (2019), 8815-8848
MSC (2010): Primary 05E15, 20F36, 20F55, 20J06
DOI: https://doi.org/10.1090/tran/7885
Published electronically: September 6, 2019
MathSciNet review: 4029713
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Abstract: We define geodesic normal forms for the general series of complex reflection groups $ G(e,e,n)$. This requires the elaboration of a combinatorial technique in order to explicitly determine minimal word representatives of the elements of $ G(e,e,n)$ over the generating set of the presentation of Corran-Picantin. Using these geodesic normal forms, we construct intervals in $ G(e,e,n)$ that are proven to be lattices. This gives rise to interval Garside groups. We determine which of these groups are isomorphic to the complex braid group $ B(e,e,n)$ and get a complete classification. For the other Garside groups that appear in our construction, we provide some of their properties and compute their second integral homology groups in order to understand these new structures.


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Additional Information

Georges Neaime
Affiliation: Fakultät für Mathematik Universität Bielefeld Universitatsstrasse 25, 33615 Brilefeld, Germany
Email: gneaime@math.uni-bielefeld.de

DOI: https://doi.org/10.1090/tran/7885
Received by editor(s): November 5, 2018
Received by editor(s) in revised form: April 10, 2019
Published electronically: September 6, 2019
Article copyright: © Copyright 2019 American Mathematical Society