Abstract: The “Bruhat decomposition” of the Grassmannian into cells is well-studied and extremely well-behaved. The common refinement of all n! permutations of it, the “matroid decomposition”, is famously awful.
Lusztig introduced a decomposition in between these for studying positive real geometry, which Postnikov recognized as the refinement of the n cyclic shifts of the Bruhat. I’ll explain how it’s just as good as the Bruhat and in some ways better, and how it naturally arises from characteristic p geometry and from Poisson geometry; also I’ll index its strata by “bounded juggling patterns”. Finally I’ll talk about recent results on the flag manifold. This work is joint with Thomas Lam and David Speyer.