Degrees of symmetric Grothendieck polynomials and Castelnuovo-Mumford regularity

Rajchgot, Jenna; Ren, Yi; Robichaux, Colleen; St. Dizier, Avery; Weigandt, Anna

doi:10.1090/proc/15294

Degrees of symmetric Grothendieck polynomials and Castelnuovo-Mumford regularity
HTML articles powered by AMS MathViewer

by Jenna Rajchgot, Yi Ren, Colleen Robichaux, Avery St. Dizier and Anna Weigandt PDF

Proc. Amer. Math. Soc. 149 (2021), 1405-1416 Request permission

Abstract:

We give an explicit formula for the degree of the Grothendieck polynomial of a Grassmannian permutation and a closely related formula for the Castelnuovo-Mumford regularity of the Schubert determinantal ideal of a Grassmannian permutation. We then provide a counterexample to a conjecture of Kummini-Lakshmibai-Sastry-Seshadri on a formula for regularities of standard open patches of particular Grassmannian Schubert varieties and show that our work gives rise to an alternate explicit formula in these cases. We end with a new conjecture on the regularities of standard open patches of arbitrary Grassmannian Schubert varieties.

References

Bruno Benedetti and Matteo Varbaro, On the dual graphs of Cohen-Macaulay algebras, Int. Math. Res. Not. IMRN 17 (2015), 8085–8115. MR 3404010, DOI 10.1093/imrn/rnu191
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
Sergey Fomin and Anatol N. Kirillov, 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 2307216
William Fulton, Flags, Schubert polynomials, degeneracy loci, and determinantal formulas, Duke Math. J. 65 (1992), no. 3, 381–420. MR 1154177, DOI 10.1215/S0012-7094-92-06516-1
D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry, available at http://www.math.uiuc.edu/Macaulay2/.
Allen Knutson, Frobenius splitting, point-counting, and degeneration, arXiv:0911.4941, 2009.
Allen Knutson and Ezra Miller, Gröbner geometry of Schubert polynomials, Ann. of Math. (2) 161 (2005), no. 3, 1245–1318. MR 2180402, DOI 10.4007/annals.2005.161.1245
Allen Knutson and Ezra Miller, Subword complexes in Coxeter groups, Adv. Math. 184 (2004), no. 1, 161–176. MR 2047852, DOI 10.1016/S0001-8708(03)00142-7
Manoj Kummini, Venkatramani Lakshmibai, Pramathanath Sastry, and C. S. Seshadri, Free resolutions of some Schubert singularities, Pacific J. Math. 279 (2015), no. 1-2, 299–328. MR 3437780, DOI 10.2140/pjm.2015.279.299
Alain Lascoux, Syzygies des variétés déterminantales, Adv. in Math. 30 (1978), no. 3, 202–237 (French). MR 520233, DOI 10.1016/0001-8708(78)90037-3
Alain Lascoux and Marcel-Paul Schützenberger, 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. 295 (1982), no. 11, 629–633 (French, with English summary). MR 686357
Cristian Lenart, Combinatorial aspects of the $K$-theory of Grassmannians, Ann. Comb. 4 (2000), no. 1, 67–82. MR 1763950, DOI 10.1007/PL00001276
Laurent Manivel, Symmetric functions, Schubert polynomials and degeneracy loci, SMF/AMS Texts and Monographs, vol. 6, American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2001. Translated from the 1998 French original by John R. Swallow; Cours Spécialisés [Specialized Courses], 3. MR 1852463
Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227, Springer-Verlag, New York, 2005. MR 2110098
Jerzy Weyman, Cohomology of vector bundles and syzygies, Cambridge Tracts in Mathematics, vol. 149, Cambridge University Press, Cambridge, 2003. MR 1988690, DOI 10.1017/CBO9780511546556
Alexander Woo and Alexander Yong, A Gröbner basis for Kazhdan-Lusztig ideals, Amer. J. Math. 134 (2012), no. 4, 1089–1137. MR 2956258, DOI 10.1353/ajm.2012.0031