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Puzzles compute (equivariant) Schubert calculus on GrassmaniansGeometry/Topology
|Speaker:||Allen Knutson, UC Berkeley|
|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.