Shadows in Coxeter Groups
The article "Shadows in Coxeter groups" by Marius Graeber and Petra Schwer is appearing in the Annals of Combinatoric (in open access!).
Links to the paper: journal, arXiv.
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 Σ. We generalize this notion and consider sets of end-alcoves of galleries that are positively folded with respect to certain orientation ϕ of Σ. 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.