The University of Michigan Combinatorics Seminar
Fall 2010
November 19, 4:10-5:00, 3866 East Hall

Puzzles for computing Schubert calculus -- some new results

Allen Knutson

Cornell University


Several years ago Terry Tao and I introduced a Littlewood-Richardson rule based on counting "puzzles", and gave a rather unsatisfying proof that it worked. In this talk I'll connect puzzles very directly to Vakil's geometric Littlewood-Richardson rule, showing in particular that all the Grassmannian subschemes in his construction are reduced and Cohen-Macaulay. With some new puzzle pieces, we can extend Vakil's work to equivariant K-theory of Grassmannians.

Schubert calculus on more general flag manifolds has been notoriously difficult to analyze combinatorially, in no small part because of its seemingly weaker connections to other mathematics. Belkale-Kumar and Ressayre showed that a certain degeneration of flag manifold Schubert calculus is relevant for "Horn's problem" questions. As I will explain, that simpler product can be computed using some other new puzzle pieces. This project is joint with Kevin Purbhoo.