
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 in a Coxeter group the elements smaller than in Bruhat order can be seen as the end-alcoves of stammering galleries of type in the Coxeter complex . We generalize this notion and consider sets of end-alcoves of galleries that are positively folded with respect to certain orientation of . 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.
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