RESEARCH GROUP

GEOMETRY

The preprint "Affine Deligne-Lusztig varieties and folded galleries governed by chimneys" by Elizabeth Milićević, Petra Schwer and Anne Thomas has recently been uploaded to the arXiv!

Link to the paper: arXiv:2006.16288.

Abstract:
We characterize the nonemptiness and dimension problems for an affine Deligne-Lusztig variety \(X_x(b)\) in the affine flag variety in terms of galleries that are positively folded with respect to a chimney. If the parabolic subgroup associated to the Newton point of \(b\) has rank 1, we then prove nonemptiness for a certain class of Iwahori-Weyl group elements \(x\) by explicitly constructing such galleries.

The preprint "Polyhedral compactifications, I" by Corina Ciobotaru, Linus Kramer and Petra Schwer was recently updated on the arXiv.

Link to the paper: arXiv.

Abstract:
In this work we describe horofunction compactifications of metric spaces and finite dimensional real vector spaces through asymmetric metrics and asymmetric polyhedral norms by means of nonstandard methods, that is, ultrapowers of the spaces at hand. The compactifications of the vector spaces carry the structure of stratified spaces with the strata indexed by dual faces of the polyhedral unit ball. Explicit neighborhood bases and descriptions of the horofunctions are provided.

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.