Colloquium Speaker:  Allen Knutson, Cornell University

"Pipe dreams, Schubert varieties, and the commuting scheme"

Schubert considered the space of kxn matrices whose Gaussian elimination has fixed pivot columns. The "volume'' of this space, in some sense, is a Schur polynomial, with many combinatorial interpretations. Pipe dreams were introduced in 1993 in [Bergeron-Billey] to give a pictorial calculus for "Schubert polynomials'', the corresponding volumes of a more general class of Schubert varieties. In 2005 Miller and I gave a geometric retrodiction of pipe dreams, based on Gröbner degeneration. In the same year I introduced the "lower-upper scheme'' {(X,Y): XY lower triangular, YX upper} in order to study the scheme of pairs of commuting matrices. I'll explain a (much more natural) pipe dream theory for the lower-upper scheme, use it to rederive the old one (also Lam-Lee-Shimozono's "bumpless pipe dreams''), and give a formula for the degree of the commuting scheme. This is joint with Paul Zinn-Justin.

This colloquium talk is part of the KOI Combinatorics Lectures.

It is supported in part by the National Science Foundation.