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} 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.

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