Shape and class of Bruhat Intervals

TL;DR

This paper explores the geometric structure of Bruhat intervals in affine Weyl groups, revealing their relation to permutohedra and proposing conjectures about their isomorphisms, with partial proofs in specific types.

Contribution

It introduces a geometric perspective on Bruhat intervals, proves isomorphism conjectures in certain cases, and uncovers that small portions encode most of the interval's information.

Findings

Bruhat intervals in type Ã_2 are generalized permutohedra minus star-shaped polygons.

Isomorphisms between Bruhat intervals may be realized by piecewise isometries.

Much information of a Bruhat interval is contained in a small part of it.

Abstract

We study Bruhat intervals in affine Weyl groups by viewing them as regions of alcoves. In type $\widetilde{A}_2$ 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 $\widetilde{A}_n$. 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 $\widetilde{A}_2$ and obtain partial results in $\widetilde{A}_n$. 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.