Shape and class of Bruhat Intervals

Abstract

We study Bruhat intervals in affine Weyl groups by viewing them as regions of alcoves. In type ${\tilde{A}}_2$ we show that each interval coincides with a generalized permutohedron minus a star-shaped polygon, and we prove a subtler version inside the dominant chamber of type ${\tilde{A}}_n$. Motivated by this geometry, we conjecture that whenever two Bruhat intervals are isomorphic, there exists an isomorphism realized by a piecewise isometry. We prove this when both endpoints are dominant in ${\tilde{A}}_2$ and obtain partial results in ${\tilde{A}}_n$. In the course of proving these results, we made the surprising observation that much of the information contained in a Bruhat interval is already encoded in a tiny portion of it.

69 pages, 25 pictures, 1 table