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The 2-braid group and Garside normal form

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Abstract

We investigate the relation between the Garside normal form for positive braids and the 2-braid group defined by Rouquier. Inspired by work of Brav and Thomas we show that the Garside normal form is encoded in the action of the 2-braid group on a certain categorified left cell module. This allows us to deduce the faithfulness of the 2-braid group in finite type. We also give a new proof of Paris’ theorem that the canonical map from the generalized braid monoid to its braid group is injective in arbitrary type.

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Acknowledgments

I would like to thank my advisor, Geordie Williamson, for his support and encouragement. I am grateful to Hanno Becker for very valuable discussions.

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  1. Max Planck Institute for Mathematics, Vivatsgasse 7, 53111, Bonn, Germany

    Lars Thorge Jensen

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  1. Lars Thorge Jensen
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Correspondence to Lars Thorge Jensen.

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Jensen, L.T. The 2-braid group and Garside normal form. Math. Z. 286, 491–520 (2017). https://doi.org/10.1007/s00209-016-1769-8

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  • DOI: https://doi.org/10.1007/s00209-016-1769-8

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