The 2-braid group and Garside normal form

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.

40 pages

References in corpus (1)

• The two-color Soergel calculus

Cited by in corpus (1)

• Mixed modular perverse sheaves on moment graphs