Gröbner geometry of formulae for Schubert polynomials
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)math.mit.edu
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