Braid Groups of Type ADE, Garside Monoids, and the categorified Root Lattice

We study Artin-Tits braid groups B-W of type ADE via the action of B-W on the homotopy category K of graded projective zigzag modules (which categorifies the action of theWeyl group W on the root lattice). Following Brav-Thomas [10], we define a metric on B-W induced by the canonical t -structure on K, and prove that this metric on B-W agrees with the word-length metric in the canonical generators of the standard positive monoid B-W(+) of the braid group. We also define, for each choice of a Coxeter element c in W, a baric structure on K.We use these baric structures to define metrics on the braid group, and we identify these metrics with the word-length metrics in the BirmanKo-Lee/Bessis dual generators of the associated dual positive monoid B-W.c(V). As consequences, we give new proofs that the standard and dual positive monoids inject into the group, give linear-algebraic solutions to the membership problem in the standard and dual positive monoids, and provide new proofs of the faithfulness of the action of B-W on K. Finally, we use the compatibility of the baric and t -structures on K to prove a conjecture of Digne and Gobet regarding the canonical word-length of the dual simple generators of ADE braid groups.