Next week, I am teaching a mini-course, introducing some of my fellow UM grad students to Schubert calculus. It will focus on the Grassmannian. Ultimately, I also hope to have all the notes from the course posted on this blog.
My goal is to help both my combinatorially- and geometrically-oriented friends learn enough of both toolsets to carry out concrete computations in intersection theory.
With these two perspectives in mind, my plan is to start with the basic geometry of the Grassmannian: Plücker and affine coordinates; line and vector bundles over ; maybe the functor of points perspective, if there’s time. I hope to do this in one day (which might turn out to be too ambitious).
Next, I’ll cover the Schubert stratification and how it describes the intersection theory / cohomology of the Grassmannian. At this point it will be natural to start talking about combinatorics: partitions and tableaux, and the ‘model’ of the cohomology ring using symmetric polynomials. This will take two days. I will state the Pieri rule, but I might skip the Jacobi-Trudi / Giambelli formula (or only include it as a remark that it implies that the Pieri rule “determines” all the other products).
Next, I will let the geometry fade into the background and focus entirely on combinatorics. It would be hopeless to prove the Littlewood-Richardson rule, so I will just describe without proof some of the many entertaining algorithms for computing it: Knutson-Tao puzzles for the entertainment value, growth diagrams for the connections to K-theory (see below), maybe Yamanouchi words. This will take a day.
Finally, with whatever time I have left, I’ll talk about some modern generalizations: Thomas and Yong’s K-theoretic growth diagrams via increasing tableaux; flag varieties and Schubert polynomials – mentioning that no positive “rule” is known, even though “positivity” itself is!
Even now I can see that this will be too much to pack into one week – optimistically, I would need 1.5-2 days for geometry, 2.5-3 days of combinatorics, and a day of generalizations. So I will begin writing lecture notes now, both to refresh my memory and to see how to make the most of the time.