Gröbner geometry of formulae for Schubert polynomials

Ezra Miller


September 29,
refreshments at 3:45pm


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)

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