An official website of the United States government
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 ofarbitraryGrassmannian Schubert varieties.
more »
« less
Castelnuovo-Mumford regularity of matrix Schubert varieties
Matrix Schubert varieties are affine varieties arising in the Schubert calculus of the complete flag variety. We give a formula for the Castelnuovo–Mumford regularity of matrix Schubert varieties, answering a question of Jenna Rajchgot. We follow her proposed strategy of studying the highest-degree homogeneous parts of Grothendieck polynomials, which we call Castelnuovo–Mumford polynomials. In addition to the regularity formula, we obtain formulas for the degrees of all Castelnuovo–Mumford polynomials and for their leading terms, as well as a complete description of when two Castelnuovo–Mumford polynomials agree up to scalar multiple. The degree of the Grothendieck polynomial is a new permutation statistic which we call the Rajchgot index; we develop the properties of Rajchgot index and relate it to major index and to weak order.
more »
« less
- PAR ID:
- 10521261
- Publisher / Repository:
- Springer Nature
- Date Published:
- Journal Name:
- Selecta Mathematica
- Volume:
- 30
- ISSN:
- 1022-1824
- Page Range / eLocation ID:
- 66
- Subject(s) / Keyword(s):
- Castelnuovo–Mumford regularity Matrix Schubert variety Grothendieck polynomial Major index Weak Bruhat order Rajchgot index
- Format(s):
- Medium: X Other: pdf
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
null (Ed.)Abstract We investigate the long-standing problem of finding a combinatorial rule for the Schubert structure constants in the $$K$$-theory of flag varieties (in type $$A$$). The Grothendieck polynomials of A. Lascoux–M.-P. Schützenberger (1982) serve as polynomial representatives for $$K$$-theoretic Schubert classes; however no positive rule for their multiplication is known in general. We contribute a new basis for polynomials (in $$n$$ variables) which we call glide polynomials, and give a positive combinatorial formula for the expansion of a Grothendieck polynomial in this basis. We then provide a positive combinatorial Littlewood–Richardson rule for expanding a product of Grothendieck polynomials in the glide basis. Our techniques easily extend to the $$\beta$$-Grothendieck polynomials of S. Fomin–A. Kirillov (1994), representing classes in connective $$K$$-theory, and we state our results in this more general context.more » « less
-
We prove a sharp lower bound on the number of terms in an element of the reduced Gröbner basis of a Schubert determinantal ideal under the term order of Knutson–Miller [Ann. of Math. (2) 161 (2005), pp. 1245–1318]. We give three applications. First, we give a pattern-avoidance characterization of the matrix Schubert varieties whose defining ideals are binomial. This complements a result of Escobar–Mészáros [Proc. Amer. Math. Soc. 144 (2016), pp. 5081–5096] on matrix Schubert varieties that are toric with respect to their natural torus action. Second, we give a combinatorial proof that the recent formulas of Rajchgot–Robichaux–Weigandt [J. Algebra 617 (2023), pp. 160–191] and Almousa–Dochtermann–Smith [Preprint, arXiv:2209.09851, 2022] computing the Castelnuovo–Mumford regularity of vexillary and toric edge ideals of bipartite graphs respectively agree for binomial . Third, we demonstrate that the Gröbner basis for given by the minimal generators of Gao–Yong [J. Commut. Algebra 16 (2024), pp. 267–273] is reduced if and only if the defining permutation is vexillary.more » « less
-
Abstract We compute the Euler characteristic of the structure sheaf of the Brill–Noether locus of linear series with special vanishing at up to two marked points. When the Brill–Noether number $$\rho $$ is zero, we recover the Castelnuovo formula for the number of special linear series on a general curve; when $$\rho =1$$, we recover the formulas of Eisenbud-Harris, Pirola, and Chan–Martín–Pflueger–Teixidor for the arithmetic genus of a Brill–Noether curve of special divisors. These computations are obtained as applications of a new determinantal formula for the $$K$$-theory class of certain degeneracy loci. Our degeneracy locus formula also specializes to new determinantal expressions for the double Grothendieck polynomials corresponding to 321-avoiding permutations and gives double versions of the flagged skew Grothendieck polynomials recently introduced by Matsumura. Our result extends the formula of Billey–Jockusch–Stanley expressing Schubert polynomials for 321-avoiding permutations as generating functions for flagged skew tableaux.more » « less
-
Abstract In this paper, we investigate the degree ofh-polynomials of edge ideals of finite simple graphs. In particular, we provide combinatorial formulas for the degree of theh-polynomial for various fundamental classes of graphs such as paths, cycles, and bipartite graphs. To the best of our knowledge, this study represents the first investigation into the combinatorial interpretation of this algebraic invariant. Additionally, we characterize all connected graphs in which the sum of the Castelnuovo–Mumford regularity and the degree of theh-polynomial of an edge ideal achieve its maximum value, equal to the number of vertices in the graph.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s00029-024-00959-x
