Gröbner geometry of formulae for Schubert polynomials

Ezra Miller

MIT

September 29,
4:15pm
refreshments at 3:45pm
2-338

ABSTRACT 

Schubert polynomials represent cohomology classes of certain subvarieties of flag manifolds. Results of Billey-Jockusch-Stanley, Fomin-Kirillov, and Bergeron-Billey show why these polynomials have nonnegative integer coefficients, by producing combinatorial formulae for them involving ``rc-graphs''. These are subsets of [n]x[n] defined in terms of reduced expressions for permutations. In this talk, I'll explain how rc-graphs represent coordinate subspaces in the vector space of n x n matrices, by showing that the determinants defining Schubert rank conditions are a Gröbner basis. This gives a new geometric proof of positivity, and produces as a corollary simple expressions for double Schubert polynomials. The proof is based on ordinary and equivariant K-theory of flag manifolds, via properties of Grothendieck polynomials. This is joint work with Allen Knutson.


Speaker's Contact Info: ezra(at-sign)math.mit.edu


Return to seminar home page

Combinatorics Seminar, Mathematics Department, MIT, sara(at-sign)math.mit.edu

Page loaded on August 31, 2000 at 10:18 PM. Copyright © 1998-99, Sara C. Billey. All rights reserved.