UW Combinatorics and Geometry Seminar: Joseph Fluegemann

We consider the Grassmannian $G{r}_{\mathbb{C}}(k,n)$, a manifold parameterized by $k$-dimensional planes in complex $n$-dimensional space. The Schubert decomposition of the Grassmannian is perhaps the classic way to split this space into smaller pieces. Positroid varieties provide a finer stratification of this space, combinatorially indexed by a special type of permutation called a bounded affine permutation. We can draw these bounded affine permutations using diagrams called affine pipe dreams. We will show that the number of such diagrams is actually connected to the question of whether certain points of these positroid varieties are smooth or singular. The proof uses the AJS/Billey formula. We further give a characterization of an affine pipe dream that corresponds to a smooth point, and this characterization can also be applied to study smoothness of Schubert varieties. In fact, if we consider the pairs (Positroid Variety, Point), we can put a partial ordering on all the pairs, and study smoothness in the context of this ordering.