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 2braid 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. 2000 Mathematics subject classification. Primary 20F36, 20J05Section 5 After introducing the important notion of an anchor we prove our main results. AcknowledgementI would like to thank my advisor, Geordie Williamson, for his support and encouragement. I am grateful to Hanno Becker for very valuable discussions. The Hecke algebra and cellsLet (W, S) be a Coxeter system, i.e. W is a group together with a set of generators S admitting a particularly nice presentation: W = ⟨s ∈ S sts . . . ms,t terms = tst . . . ms,t terms , s 2 = 1⟩ where m s,t ⩾ 2 is the order of st for s ≠ t ∈ S. For w ∈ W denote the left (resp. right) descent set of w by L(w) = {s ∈ S sw < w} (resp. R(w) = {s ∈ S ws < w}). Let H (W,S) be the corresponding Hecke algebra over Z[v, v −1 ] which we will also denote by H if there is no danger of confusion. Denote by {H w } w∈W the standard and by {H w } w∈W the Kazhdan-Lusztig basis in Soergel's normalization (see [Soe97]). Since we are not working with the usual normalization, let us state the relations for the standard basis: H 2 s = (v −1 − v)H s + 1 for all s ∈ S, H s H t H s . . . ms,t terms = H t H s H t . . . ms,t terms for all s ≠ t ∈ S.

Soergel Calculus with patches

“…(where B W is considered as a category with only the identity maps) which Rouquier conjectured to be faithful. This was shown in type A in [17], in simply laced finite type in [4], and in all finite types in [13].…”

Dual Garside structures and Coxeter sortable elements

“…for all s, t ∈ S. As this presentation is positive, let B(W ) + denote the monoid generated by the same presentation as B(W ). It is a general fact that B(W ) + ⊆ B(W ) (see [35], [33]). If W is finite, the group B(W ) (or W ) is said to be of spherical type.…”

Twisted filtrations of Soergel bimodules and linear Rouquier complexes

“…Here η s (r) = 1 2 (r ⊗ f s +rf s ⊗1) for any r ∈ R and µ s is the multiplication map. The functors F s ⊗ − and E s ⊗ − define mutually inverse equivalences of K b (R), categorifying the braid group of the Coxeter system in finite type (see [26], [25], [27], [19] and the references therein). Given x ∈ W with reduced expression st • • • u, we write F x for the Rouquier complex corresponding to the positive canonical lift x of x in the braid group, that is,…”