Diagonal degenerations of matrix Schubert varieties

TL;DR

This paper proves a conjecture linking diagonal degenerations of matrix Schubert varieties to bumpless pipe dreams, expanding understanding of their algebraic and combinatorial structures using liaison and geometric vertex decomposition techniques.

Contribution

It provides a complete proof of the conjecture relating diagonal degenerations and bumpless pipe dreams for all permutations, utilizing liaison and geometric vertex decomposition methods.

Findings

Proved the conjecture in full generality.

Connected diagonal degenerations with bumpless pipe dreams.

Enhanced understanding of matrix Schubert varieties' algebraic structure.

Abstract

Knutson and Miller (2005) established a connection between the anti-diagonal Gr\"obner 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. Recently, Hamaker, Pechenik, and Weigandt (2022) proposed a similar connection between diagonal degenerations of matrix Schubert varieties and bumpless pipe dreams, newer combinatorial objects introduced by Lam, Lee, and Shimozono (2021). Hamaker, Pechenik, and Weigandt described new generating sets of the defining ideals of matrix Schubert varieties and conjectured a characterization of permutations for which these generating sets form diagonal Gr\"obner bases. They proved special cases of this conjecture and described diagonal degenerations of matrix Schubert varieties in terms of bumpless pipe dreams in these cases. The purpose of this paper is to prove the conjecture in full generality. The proof uses a connection between liaison and geometric vertex decomposition established in earlier work with Rajchgot (2021).