Diagrammatic Calculus of Coxeter and Braid Groups

Abstract

We investigate a novel diagrammatic approach to examining strict actions of a Coxeter group or a braid group on a category. This diagrammatic language, which was developed in a series of papers by Elias, Khovanov and Williamson, provides new tools and methods to attack many problems of current interest in representation theory. In our research we considered a particular problem which arises in this context. To a Coxeter group $W$ one can associate a real hyperplane arrangement, and can consider the complement of these hyperplanes in the complexification $Y_W$. The celebrated $K(\pi,1)$ conjecture states that $Y_W$ should be a classifying space for the pure braid group, and thus a natural quotient ${Y_W}/{W}$ should be a classifying space for the braid group. Salvetti provided a cell complex realization of the quotient, which we refer to as the Salvetti complex. In this paper we investigate a part of the $K(\pi,1)$ conjecture, which we call the $K(\pi,1)$ conjecturette, that states that the second homotopy group of the Salvetti complex is trivial. In this paper we present a diagrammatic proof of the $K(\pi,1)$ conjecturette for a family of braid groups as well as an analogous result for several families of Coxeter groups.