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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