Geodesic
Diagonal degenerations of matrix Schubert varieties
Klein, Patricia1
1 Texas A&M University Department of mathematics College Station TX 77843 (USA)
Algebraic Combinatorics, Tome 6 (2023) no. 4, pp. 1073-1094

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Knutson and Miller (2005) 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. 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 Gröbner 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).

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DOI : 10.5802/alco.296
Classification : 14M15, 13P10, 13C40, 05E99
Keywords: Matrix Schubert varieties, Gröbner bases, Gorenstein liaison, geometric vertex decomposition

Klein, Patricia 1

1 Texas A&M University Department of mathematics College Station TX 77843 (USA)
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Klein, Patricia. Diagonal degenerations of matrix Schubert varieties. Algebraic Combinatorics, Tome 6 (2023) no. 4, pp. 1073-1094. doi: 10.5802/alco.296

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