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Puzzles compute (equivariant) Schubert calculus on GrassmaniansGeometry/Topology
|Speaker: ||Allen Knutson, UC Berkeley|
|Location: ||593 Kerr|
|Start time: ||Thu, Oct 18 2001, 3:10PM|
The cell decomposition of the complex Grassmannian Gr_k(C^n) gives a
basis for cohomology, and there are famous formulae such as
Littlewood-Richardson to compute the (nonnegative!) structure
constants for multiplication in this basis.
For example, (S_0101)^4 = 2 S_1100 is the "through every four generic
lines in space, pass two others" calculation. Unfortunately these
formulae hide most of the symmetries of the problem.
We introduce a new scheme, in terms of counting "puzzles", in which most
of these symmetries are manifest. In fact they can be made to compute
the torus-equivariant cohomology as well (which we will introduce
from scratch). This allows for an inductive proof starting from
the "most equivariant" case. This is joint work with Terry Tao of UCLA.