A Weighted Enumeration of Maximal Chains in the Bruhat Order

Abstract

Given a finite Weyl group W with root system Φ, assign the weight α ∈ Φ to each covering pair in the Bruhat order related by the reflection corresponding to α. Extending this multiplicatively to chains, we prove that the sum of the weights of all maximal chains in the Bruhat order has an explicit product formula, and prove a similar result for a weighted sum over maximal chains in the Bruhat ordering of any parabolic quotient of W. Several variations and open problems are discussed.