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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 |
Description
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.