The parabolic coset structure of Bruhat intervals in Coxeter groups

Abstract

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 Poincaré polynomial of a Coxeter group element. We also show that the fibers of standard parabolic projection maps on Schubert varieties are themselves Schubert varieties.

11 pages, 2 figures. Ver2: Updated Section 2 with more background. Updated example at the end of Section 3