Generators and Relations for Bott–Samelson Bimodules and the Double Leaves Basis

Abstract

In the previous chapters we described a diagrammatic category \({\mathcal {H}_{\mathrm {BS}}}\) associated to dihedral Coxeter systems, and a functor to the category \(\mathbb {B}\mathbb {S}\mathrm {Bim}\) of Bott–Samelson bimodules (which is meant to be an equivalence). To extend this definition to an arbitrary Coxeter system, it turns out that the only new relations needed are associated with finite rank 3 parabolic subgroups. In this chapter we describe the category \({\mathcal {H}_{\mathrm {BS}}}\) for an arbitrary Coxeter system, discussing the new details and giving a different set of motivations. We also briefly discuss what happens for other realizations, where the diagrammatic category may behave better than the algebraic category \(\mathbb {B}\mathbb {S}\mathrm {Bim}\).