Partial orders on the Weyl group, monoids in the braid group, and homological algebra

Let W be a finite Weyl group. Associated to W there are two important partial orders - (weak) Bruhat order, and absolute order. These two partial orders are related to a pair of combinatorial lattices, and these lattices are in some (not completely understood) senses "dual" to one another. As a result many other structures related to W come in dual pairs; for example, as we will explain, the braid group of W has a pair of "dual" Garside structures, and a pair of "dual" positive monoids. The goal of this talk will be to explain how these dual structures appear in the higher representation theory of the braid group. (Joint with Hoel Queffelec.) ; Non UBC ; Unreviewed ; Author affiliation: Australian National University ; Faculty