Diagonal degenerations of matrix Schubert varieties

Klein, Patricia

doi:10.5802/alco.296

Klein, Patricia ¹

¹ Texas A&M University Department of mathematics College Station TX 77843 (USA)

Algebraic Combinatorics, Volume 6 (2023) no. 4, pp. 1073-1094.

Abstract

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).

Received: 2020-09-08
Revised: 2023-01-19
Accepted: 2023-01-21
Published online: 2023-08-29

DOI: 10.5802/alco.296

Classification: 14M15, 13P10, 13C40, 05E99
Keywords: Matrix Schubert varieties, Gröbner bases, Gorenstein liaison, geometric vertex decomposition

Author's affiliations:

Klein, Patricia ¹

¹ Texas A&M University Department of mathematics College Station TX 77843 (USA)

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{ALCO_2023__6_4_1073_0,
     author = {Klein, Patricia},
     title = {Diagonal degenerations of matrix {Schubert} varieties},
     journal = {Algebraic Combinatorics},
     pages = {1073--1094},
     publisher = {The Combinatorics Consortium},
     volume = {6},
     number = {4},
     year = {2023},
     doi = {10.5802/alco.296},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.296/}
}

TY  - JOUR
AU  - Klein, Patricia
TI  - Diagonal degenerations of matrix Schubert varieties
JO  - Algebraic Combinatorics
PY  - 2023
SP  - 1073
EP  - 1094
VL  - 6
IS  - 4
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.296/
DO  - 10.5802/alco.296
LA  - en
ID  - ALCO_2023__6_4_1073_0
ER  -

%0 Journal Article
%A Klein, Patricia
%T Diagonal degenerations of matrix Schubert varieties
%J Algebraic Combinatorics
%D 2023
%P 1073-1094
%V 6
%N 4
%I The Combinatorics Consortium
%U https://alco.centre-mersenne.org/articles/10.5802/alco.296/
%R 10.5802/alco.296
%G en
%F ALCO_2023__6_4_1073_0

Klein, Patricia. Diagonal degenerations of matrix Schubert varieties. Algebraic Combinatorics, Volume 6 (2023) no. 4, pp. 1073-1094. doi : 10.5802/alco.296. https://alco.centre-mersenne.org/articles/10.5802/alco.296/

References
Cited by

[1] Abe, Hiraku; Billey, Sara Consequences of the Lakshmibai-Sandhya theorem: the ubiquity of permutation patterns in Schubert calculus and related geometry, Schubert calculus—Osaka 2012 (Adv. Stud. Pure Math.), Volume 71, Math. Soc. Japan, [Tokyo], 2016, pp. 1-52 | DOI | MR | Zbl

[2] Abhyankar, Shreeram S. Enumerative combinatorics of Young tableaux, Monographs and Textbooks in Pure and Applied Mathematics, 115, Marcel Dekker, Inc., New York, 1988, xx+509 pages | MR

[3] Bergeron, Nantel; Billey, Sara RC-graphs and Schubert polynomials, Experiment. Math., Volume 2 (1993) no. 4, pp. 257-269 | DOI | MR | Zbl

[4] Bernšteĭn, I. N.; Gel’fand, I. M.; Gel’fand, S. I. Schubert cells, and the cohomology of the spaces $G / P$ , Uspehi Mat. Nauk, Volume 28 (1973) no. 3(171), pp. 3-26 | MR | Zbl

[5] Boocher, Adam Free resolutions and sparse determinantal ideals, Math. Res. Lett., Volume 19 (2012) no. 4, pp. 805-821 | DOI | MR | Zbl

[6] Bruns, Winfried Algebras defined by powers of determinantal ideals, J. Algebra, Volume 142 (1991) no. 1, pp. 150-163 | DOI | MR | Zbl

[7] Bruns, Winfried; Conca, Aldo KRS and powers of determinantal ideals, Compositio Math., Volume 111 (1998) no. 1, pp. 111-122 | DOI | MR | Zbl

[8] Bruns, Winfried; Römer, Tim; Wiebe, Attila Initial algebras of determinantal rings, Cohen-Macaulay and Ulrich ideals, Michigan Math. J., Volume 53 (2005) no. 1, pp. 71-81 | DOI | MR | Zbl

[9] Buciumas, Valentin; Scrimshaw, Travis Double Grothendieck polynomials and colored lattice models, Int. Math. Res. Not. IMRN (2022) no. 10, pp. 7231-7258 | DOI | MR | Zbl

[10] Conca, A. Gröbner bases and determinantal rings, Ph. D. Thesis, Universitat Essen (1993)

[11] Conca, Aldo Straightening law and powers of determinantal ideals of Hankel matrices, Adv. Math., Volume 138 (1998) no. 2, pp. 263-292 | DOI | MR | Zbl

[12] Conca, Aldo; De Negri, Emanuela; Gorla, Elisa Universal Gröbner bases for maximal minors, Int. Math. Res. Not. IMRN (2015) no. 11, pp. 3245-3262 | DOI | MR | Zbl

[13] Conca, Aldo; Herzog, Jürgen Ladder determinantal rings have rational singularities, Adv. Math., Volume 132 (1997) no. 1, pp. 120-147 | DOI | MR | Zbl

[14] De Negri, Emanuela; Sbarra, Enrico Gröbner bases of ideals cogenerated by Pfaffians, J. Pure Appl. Algebra, Volume 215 (2011) no. 5, pp. 812-821 | DOI | MR | Zbl

[15] Eisenbud, David Commutative algebra. With a view toward algebraic geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, New York, 1995, xvi+785 pages | MR

[16] Fink, Alex; Mészáros, Karola; St. Dizier, Avery Zero-one Schubert polynomials, Math. Z., Volume 297 (2021) no. 3-4, pp. 1023-1042 | DOI | MR | Zbl

[17] Fomin, Sergey; Kirillov, Anatol N. Grothendieck polynomials and the Yang-Baxter equation, Formal power series and algebraic combinatorics/Séries formelles et combinatoire algébrique, DIMACS, Piscataway, NJ, sd, pp. 183-189 | MR

[18] Fulton, William Flags, Schubert polynomials, degeneracy loci, and determinantal formulas, Duke Math. J., Volume 65 (1992) no. 3, pp. 381-420 | DOI | MR | Zbl

[19] Fulton, William Young tableaux: With applications to representation theory and geometry, London Mathematical Society Student Texts, Cambridge University Press, Cambridge, 1996 | DOI

[20] Gonciulea, Nicolae; Miller, Claudia Mixed ladder determinantal varieties, J. Algebra, Volume 231 (2000) no. 1, pp. 104-137 | DOI | MR

[21] Gorla, E.; Migliore, J. C.; Nagel, U. Gröbner bases via linkage, J. Algebra, Volume 384 (2013), pp. 110-134 | DOI | MR | Zbl

[22] Gorla, Elisa Mixed ladder determinantal varieties from two-sided ladders, J. Pure Appl. Algebra, Volume 211 (2007) no. 2, pp. 433-444 | DOI | MR | Zbl

[23] Gorla, Elisa A generalized Gaeta’s theorem, Compos. Math., Volume 144 (2008) no. 3, pp. 689-704 | DOI | MR | Zbl

[24] Hamaker, Zachary; Pechenik, Oliver; Weigandt, Anna Gröbner geometry of Schubert polynomials through ice, Adv. Math., Volume 398 (2022), Paper no. 108228, 29 pages | DOI | MR | Zbl

[25] Herzog, Jürgen; Trung, Ngô Viêt Gröbner bases and multiplicity of determinantal and Pfaffian ideals, Adv. Math., Volume 96 (1992) no. 1, pp. 1-37 | DOI | MR | Zbl

[26] Hochster, M.; Eagon, John A. Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci, Amer. J. Math., Volume 93 (1971), pp. 1020-1058 | DOI | MR | Zbl

[27] Klein, Patricia; Rajchgot, Jenna Geometric vertex decomposition and liaison, Forum Math. Sigma, Volume 9 (2021), Paper no. e70, 23 pages | DOI | MR | Zbl

[28] Knutson, Allen; Miller, Ezra Gröbner geometry of Schubert polynomials, Ann. of Math. (2), Volume 161 (2005) no. 3, pp. 1245-1318 | DOI | MR | Zbl

[29] Knutson, Allen; Miller, Ezra; Yong, Alexander Gröbner geometry of vertex decompositions and of flagged tableaux, J. Reine Angew. Math., Volume 630 (2009), pp. 1-31 | DOI | MR | Zbl

[30] Lakshmibai, V.; Sandhya, B. Criterion for smoothness of Schubert varieties in $Sl (n) / B$ , Proc. Indian Acad. Sci. Math. Sci., Volume 100 (1990) no. 1, pp. 45-52 | DOI | MR | Zbl

[31] Lam, Thomas; Lee, Seung Jin; Shimozono, Mark Back stable Schubert calculus, Compos. Math., Volume 157 (2021) no. 5, pp. 883-962 | DOI | MR | Zbl

[32] Lascoux, A. Chern and Yang through ice (2002) http://tinyurl.com/y64bnpro

[33] Lascoux, Alain; Schützenberger, Marcel-Paul Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck d’une variété de drapeaux, C. R. Acad. Sci. Paris Sér. I Math., Volume 295 (1982) no. 11, pp. 629-633 | MR | Zbl

[34] Narasimhan, Himanee The irreducibility of ladder determinantal varieties, J. Algebra, Volume 102 (1986) no. 1, pp. 162-185 | DOI | MR | Zbl

[35] Weigandt, Anna Bumpless pipe dreams and alternating sign matrices, J. Combin. Theory Ser. A, Volume 182 (2021), Paper no. 105470, 52 pages | DOI | MR | Zbl

[36] Woo, Alexander; Yong, Alexander When is a Schubert variety Gorenstein?, Adv. Math., Volume 207 (2006) no. 1, pp. 205-220 | DOI | MR | Zbl

Cited by Sources: