Shadows in Coxeter groups

TL;DR

This paper introduces the concept of shadows in Coxeter groups, generalizing Bruhat order end-alcoves to positively folded galleries, and explores their properties and connections to algebraic structures.

Contribution

It generalizes the notion of end-alcoves in Coxeter groups to positively folded galleries called shadows, and studies their algorithmic properties.

Findings

Defined various notions of orientations and shadows

Analyzed algorithmic properties of shadows

Connected shadows to algebraic structures like affine Deligne-Lusztig varieties

Abstract

For a given $w$ in a Coxeter group $W$ the elements $u$ smaller than $w$ in Bruhat order can be seen as the end-alcoves of stammering galleries of type $w$ in the Coxeter complex $\Sigma$. We generalize this notion and consider sets of end-alcoves of galleries that are positively folded with respect to certain orientation $\phi$ of $\Sigma$. We call these sets shadows. Positively folded galleries are closely related to the geometric study of affine Deligne-Lusztig varieties, MV polytopes, Hall-Littlewood polynomials and many more agebraic structures. In this paper we will introduce various notions of orientations and hence shadows and study some of their algorithmic properties.