Topology & Geometric Group Theory Seminar

Fall 2007

1:30 - 2:30, Malott 253

Tuesday, November 6

Allen Knutson, UC San Diego

Schubert calculus and geometric shifting: combinatorics meets combinatorics, via geometry

A classic formula in algebraic combinatorics is the Littlewood-Richardson rule, which computes in a manifestly positive way the multiplicative structure constants of the Schur function basis of the ring of symmetric functions. A classic theorem of extremal combinatorics is the Erdos-Ko-Rado theorem, which describes the maximum solutions to the following problem: consider k-subsets of {1,…,n} such that any two intersect. Then (for n>2k) the only way to maximize a collection of such is for there to exist a special element that lies in every subset. To prove their theorem, EKR invented the "shifting" technique. I'll recall its definition, and define a geometric version, which I'll use to prove a new Littlewood-Richardson rule, thought of in terms of intersection theory on Grassmannians.

This work is joint with Ravi Vakil.

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