The Geometric Formula for affine Weyl groups

Abstract

Let W be an affine Weyl group with corresponding finite Weyl group Wf . For each λ, a dominant coweight, corresponds an element θ(λ) ∈ W. With N. Libedinsky and D. Plaza, we produce a conjecture called the Geometric Formula predicting the following: the cardinality of the set of elements in W that are lesser or equal to θ(λ) in the Bruhat order, is a linear combination (with coefficients not depending on λ) of the volumes of the faces of the polytope Conv(λ), constructed as the convex hull of the set Wf · λ. We prove the geometric formula for type fA3, by giving general algebraic and geometric constructions for the set ≤ θ(λ). We study the polytope Conv(λ), its faces, and give some formulas to compute their volumes of the corresponding dimension.