On the size of Bruhat intervals

TL;DR

This paper introduces a convex geometry formula to determine the size of lower Bruhat intervals in affine Weyl groups, supported by computational evidence suggesting broader applicability.

Contribution

It provides a new convex geometry-based formula for affine Weyl group Bruhat intervals and hypothesizes its generalization to all lower Bruhat intervals.

Findings

Convex geometry formula for affine Weyl groups

Computational evidence for broader applicability

Hypothesis of a universal formula for all lower Bruhat intervals

Abstract

For affine Weyl groups and elements associated to dominant coweights, we present a convex geometry formula for the size of the corresponding lower Bruhat intervals. Extensive computer calculations for these groups have led us to believe that a similar formula exists for all lower Bruhat intervals.