Annexes in affine Coxeter complexes

TL;DR

This paper introduces the concept of annexes in affine Coxeter complexes, revealing their properties and geometric structure, and providing new insights into the Bruhat order through a novel perspective.

Contribution

It defines annexes in Coxeter groups, proves their finiteness in affine cases, and describes their geometric structure in rank-two complexes, offering a new viewpoint on Bruhat order.

Findings

Annexes are finite in affine Coxeter groups.

Geometric structure of annex boundaries characterized in rank-two cases.

Provides a new perspective on Bruhat intervals and folded galleries.

Abstract

We introduce the annex of an element $x$ in a Coxeter group as the set of elements $y$ such that $x \nleq y$ with respect to Bruhat order. This notion provides a complementary perspective to the study of Bruhat intervals and their interpretation via folded galleries. We establish general properties of annexes and show that in affine Coxeter groups the annex of any fixed element is finite. In rank-two affine Coxeter complexes, we further describe the geometric structure of annex boundaries using descent sets and configurations of parallel reflections. These results offer a new geometric viewpoint on the structure of the Bruhat order.