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The parabolic coset structure of Bruhat intervals in Coxeter groups
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In this paper, we study the decomposition of Bruhat intervals in a Coxeter group with respect to cosets of a parabolic subgroup. Our main result is that the intersection of a lower Bruhat interval with a parabolic coset contains a unique maximal element. As an application, we give a decomposition formula for the Poincar\'{e} polynomial of a Coxeter group element. We also show that the fibers of standard parabolic projection maps on Schubert varieties are themselves Schubert varieties.
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Demazure product and hopping in type D
The Demazure product on type D Coxeter groups is computed by applying a sequence of hopping operators to the ordinary product, with each operator read off from the unfolding of the second factor.
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