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Knutson and Miller established a connection between the anti-diagonal Gröbner degenerations of matrix Schubert varieties and the pre-existing combinatorics of pipe dreams. They used this correspondence to give a geometrically-natural explanation for the appearance of the combinatorially-defined Schubert polynomials as representatives of Schubert classes. The goal of this talk is to tell a similar story for diagonal degenerations and bumpless pipe dreams, newer combinatorial objects introduced by Lam, Lee, and Shimozono.
In the first half of this talk, we will review standard material on Hilbert series, Gröbner bases, and vertex decomposition of simplicial complexes. Our running examples will come from matrix Schubert varieties and (more generally) alternating sign matrix varieties, which are classes of generalized determinantal varieties. In the second half, we will describe a relationship between diagonal degenerations of these varieties and bumpless pipe dreams. Finally, we will discuss some related commutative-algebraic questions about these varieties that remain open. This talk is based on joint work with Anna Weigandt.
Bumpless Pipe Dreams Encode Gröbner Geometry of Schubert Polynomials