2017/03/17 by Anthony Licata, Licata, Anthony M., Hoel Queffélec +1 · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #Advanced Algebra and Geometry #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1703.06011
We study Artin-Tits braid groups \mathbbBW of type ADE via the action of \mathbbBW 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, we define a metric on \mathbbBW induced by the canonical t-structure on K, and prove that this metric on \mathbbBW agrees with the word-length metric in the canonical generators of the standard positive monoid \mathbbBW+ 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 \mathbbBW.c^\vee. 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 \mathbbBW 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.