Braid groups of type ADE, Garside monoids, and the categorified root lattice

Abstract

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 the Weyl group W on the root lattice). Following Brav-Thomas [BT11], 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 wordlength 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 Birman-Ko-Lee/Bessis dual generators of the associated dual positive monoid B ∨ W.c . 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.Contents 10 3. Categorical ping pong 17 4. Homological interpretations of word-length metrics 28 5. Standard and dual word-length metrics and a conjecture of Digne-Gobet 37 References 391. Artin-Tits braid groups in type ADE

Non-crossing partitions

“…Notice that Licata and Queffelec [44] have a proof of Theorem 4.4 in types A,D,E with a different approach using categorification.…”