|Date: Friday, October 17, 2014
Location: 3866 East Hall (4:10 PM to 5:00 PM)
Title: Schubert calculus and shifting of interval positroid varieties
Abstract: If X is a subvariety of a Grassmannian, then the expansion of its homology class [X] into Schubert classes has nonnegative coefficients, so for combinatorially-defined X we should hope to find combinatorial formulae for these coefficients. In the most familiar version, X is the intersection of a Schubert variety and opposite Schubert variety (a "Richardson variety"), and this expansion is given by the Littlewood-Richardson rule.
Vakil defined a larger class of "checkerboard varieties" X, and used one-parameter degenerations to give a positive recurrence relation on their homology classes, thus a "geometric Littlewood-Richardson rule" counting (2+1)-dimensional "checker games".
My first main result builds on this several ways:
* I relate his degenerations to Erd\H os-Ko-Rado shifting
* I expand his class of varieties to "interval positroid varieties"
* Using results of [K-Lam-Speyer] on those varieties, I compute not just in homology, but equivariant K-theory
* I simplify Vakil's (2+1)-dimensional checker games to more computationally convenient (1+1)-dimensional "IP pipe dreams".
Mathias Lederer and I apply this to study a very natural 2-parameter deformation of the ring Symm of symmetric functions. I'll explain how Symm can be built by applying H_* to the "direct sum" map on Grassmannians, and twice deformed by replacing H_* with equivariant K-homology. Then I'll compute the structure constants in this deformation, using a variant of IP pipe dreams.
Speaker: Allen Knutson
Institution: Cornell U.