Bruhat Order and Coxeter Hyperplane Arrangements
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In the paper “Bruhat order, rationally smooth Schubert varieties, and hyperplane arrangements,” S. Oh and H. Yoo studied Schubert varieties in generalized flag manifolds by linking them with a certain hyperplane arrangement coming from the reflection synmmetries of a Weyl group. They made two conjectures. The first relates to a curious property of maximal parabolic quotients of finite Weyl groups. The second states that for an element w of a finite Coxeter group W, the generating function R_w(q) of its hyperplane arrangement coincides with the rank-generating function P_w(q) of its lower interval [e,w] in the Bruhat order, if and only if [e, w] is rank-symmetric. Here, we generalize the first conjecture for finite Coxeter groups and prove it. We use this result to prove the second conjecture. Two chapters of background material on Coxeter systems and hyperplane arangements are provided.