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The puzzle conjecture for 2-step flag manifolds

Anders Buch
(Rutgers University)
Algebraisk geometriseminar
Onsdag, 22 august, 2012, at 14:15-15:00, in Aud. D3 (1531-215)
A conjecture of Allen Knutson from 1999 asserts that the Schubert structure constants of the cohomology ring of any flag variety SL(n)/P are equal to the number of triangular puzzles with specified border labels that can be constructed using a list of puzzle pieces.  Knutson quickly found a counter example to the general conjecture.  Joint work of myself, Kresch, and Tamvakis later showed the (3-point, genus zero) Gromov-Witten invariants of Grassmannians are special cases of the structure constants of two-step flag varieties, and we suggested that Knutson's conjecture might be true in this special case, backed up by computer verification for n <= 16.  I will speak about a proof of this conjecture, joint with Andrew Kresch, Kevin Purbhoo, and Tamvakis.  I will also explain a generalization to the equivariant structure constants of two-step flag varieties that I have recently found a proof of.