Schubert Varieties, Subword Complexes, Frobenius Splitting, and Bruhat Atlases

We consider stratifications of schemes by subvarieties, and introduce the notion of a stratification generated by a collection of (reduced) subschemes. We recall some basics of Gröbner degenerations (just for lex order) and Stanley-Reisner theory. We then state the main theorem, that the lex init of a Kazhdan-Lusztig variety in BottSamelson coördinates is the Stanley-Reisner scheme of a subword complex, after defining all those terms. Rather than getting into the combinatorial details, we focus on the principal ingredient in the proof, which is Frobenius splitting. Now that we have an interesting class of vector spaces with stratifications, we glue them together in atlases to make some famous stratified spaces, such as flag manifolds and wonderful compactifications of groups. These are notes for the Dijon summer school TLAG 2017, largely compiled from [Kn09] and [KnMi04]. Remerciements to the organizers for the invitation.