Positively folded galleries in Coxeter complexes play a role in many areas of maths, such as in the study of affine Hecke algebras, Macdonald polynomials, MV-polytopes, and affine Deligne-Lusztig varieties.
In this talk, we will define positively folded galleries, and then look at a new recursive description of the set of end alcoves of folded galleries which are positive with respect to alcove-induced orientations. This further allow us to find the images of retractions from certain points at infinity, giving us a combinatorial description of double coset intersections in the affine flag variety. This talk is based on joint work with Elizabeth Milićević, Petra Schwer and Anne Thomas.
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