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Algebraic Geometry Seminar, UC Davis

Allen Knutson, Cornell University

The commuting scheme and generic pipe dreams

Organizer's time: 2021-11-30 11:00 America/Los_Angeles

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The space of pairs of commuting matrices is more mysterious than you might think -- in particular, Hochster's 1984 conjecture that it is reduced remains unresolved. I'll explain how to degenerate it to one component of the "lower-upper scheme" {(X,Y) : XY lower triangular, YX upper triangular}, a reduced complete intersection, and how to compute the degree of any component as a sum over "generic pipe dreams". As a consequence, this recovers both the "pipe dream" and "bumpless pipe dream" formulae for double Schubert polynomials. Some of this work is joint with Paul Zinn-Justin.

Submitted by: Eugene Gorsky