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.
Universidad de Chile
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Attribution-NonCommercial-NoDerivs 3.0 United States