Classification of rigid intervals in Ã2

Abstract

Let W be an affine Weyl group of type Ã2 with corresponding finite Weyl group Wf . In this thesis, we study certain family of Bruhat intervals by exploring the relationship between the geometry of alcoves associated with the Coxeter complex of W and Euclidean geometry. First, we provide a characterization of lower intervals for each element of W in terms of convex sets, completing the characterization given in [LP23]. Next, we give a geometric description of a family intervals [x, y] which allow us to associate a polygon to its intervals. On the other hand, using the alcove geometry, we introduce a new poset isomorphism between these intervals. Finally, by utilizing the combinatorial information encoded in the polygon associated with each interval, we classify the intervals satisfying mild restrictions up to poset isomorphism.