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``Discretely ending the year II'': Schubert calculus of interval rank varieties

Algebra & Discrete Mathematics

Speaker: Prof. Allen Knutson, Cornell University
Location: 2112 MSB
Start time: Mon, Jun 9 2014, 4:10PM

If X is a subvariety of a Grassmannian, then its homology class [X] is a positive combination of the Schubert basis of the homology of the Grassmannian. The structure constants of the cohomology product (the "Littlewood-Richardson coefficients") arise this way, when X is the intersection of a Schubert variety with an opposite Schubert variety. I'll define a family of "interval rank varieties", interpolating between Schubert varieties and these intersections, and give a formula for their classes as a sum over certain diagrams of pipes. The basic inductive step is a geometric version of Erd\H os-Ko-Rado shifting. This formula builds on Vakil's "geometric Littlewood-Richardson rule" in three ways: (1) it applies to a larger class of varieties, (2) each term is given by a 2-dimensional "IP pipe dream" rather than a (2+1)-dimension "checker game", and (3) it computes in T-equivariant K-theory, not just homology.

This the second part of event. It will be followed by a reception.