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Zero-one dual characters of flagged Weyl modules

Part of: Combinatorics Algebraic combinatorics Projective and enumerative geometry

Published online by Cambridge University Press:  08 July 2025

Peter L. Guo ,
Zhuowei Lin  and
Simon C. Y. Peng
Peter L. Guo*
Affiliation:
Center for Combinatorics, Nankai University , LPMC, Tianjin 300071, P.R. China
Zhuowei Lin
Affiliation:
Center for Combinatorics, Nankai University , LPMC, Tianjin 300071, P.R. China e-mail: zwlin0825@163.com
Simon C. Y. Peng
Affiliation:
Center for Applied Mathematics, Tianjin University, Tianjin 300072, P.R. China e-mail: pcy@tju.edu.cn
*
e-mail: lguo@nankai.edu.cn
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Abstract

We prove a criterion of when the dual character $\chi _{D}(x)$ of the flagged Weyl module associated a diagram D in the grid $[n]\times [n]$ is zero-one, that is, the coefficients of monomials in $\chi _{D}(x)$ are either 0 or 1. This settles a conjecture proposed by Mészáros–St. Dizier–Tanjaya. Since Schubert polynomials and key polynomials occur as special cases of dual flagged Weyl characters, our approach provides a new and unified proof of known criteria for zero-one Schubert/key polynomials due to Fink–Mészáros–St. Dizier and Hodges–Yong, respectively.

Keywords

Flagged Weyl moduledual characterSchubert polynomialkey polynomialzero-one polynomial

MSC classification

Primary: 05E10: Combinatorial aspects of representation theory 05A19: Combinatorial identities, bijective combinatorics 14N15: Classical problems, Schubert calculus

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Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 23
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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References

Armon, S., Assaf, S., Bowling, G., and Ehrhard, H., Kohnert’s rule for flagged Schur modules . J. Algebra 617(2023), 352–381.10.1016/j.jalgebra.2022.10.032CrossRefGoogle Scholar
Assaf, S. and Searles, D., Kohnert polynomials . Exp. Math. 31(2022), no. 1, 93–119.10.1080/10586458.2019.1588180CrossRefGoogle Scholar
Assaf, S. and Searles, D., Kohnert tableaux and a lifting of quasi-Schur functions . J. Combin. Theory Ser. A 156(2018), 85–118.10.1016/j.jcta.2018.01.001CrossRefGoogle Scholar
Brion, M., Multiplicity-free subvarieties of flag varieties . In: Commutative algebra (Grenoble/Lyon, 2001), Contemporary Mathematics, 331, American Mathematical Society, Providence, RI, 2003, pp. 13–23.CrossRefGoogle Scholar
Castillo, F., Cid-Ruiz, Y., Mohammadi, F., and Montaño, J., K-polynomials of multiplicity-free varieties. Preprint, 2022. arXiv:2212.13091.Google Scholar
Eur, C. and Larson, M., K-theoretic positivity for matroids. Preprint, 2023. arXiv:2311.11996.Google Scholar
Fan, N. J. Y. and Guo, P. L., Vertices of schubitopes . J. Combin. Theory Ser. A 177(2021), 105311, 20pp.Google Scholar
Fan, N. J. Y. and Guo, P. L., Upper bounds of Schubert polynomials . Sci. China Math. 65(2022), 1319–1330.10.1007/s11425-020-1843-5CrossRefGoogle Scholar
Fink, A., Mészáros, K., and St, A., Dizier, Schubert polynomials as integer point transforms of generalized permutahedra . Adv. Math. 332(2018), 465–475.10.1016/j.aim.2018.05.028CrossRefGoogle Scholar
Fink, A., Mészáros, K., and St Dizier, A., Zero-one Schubert polynomials . Math. Z. 297(2021), 1023–1042.10.1007/s00209-020-02544-2CrossRefGoogle Scholar
Gao, Y., Hodges, R., and Yong, A., Classification of Levi-spherical Schubert varieties . Selecta Math. (N.S.) 29(2023), Paper No. 55. 40 pp.10.1007/s00029-023-00856-9CrossRefGoogle Scholar
Hamaker, Z., Pechenik, O., and Weigandt, A., Gröbner geometry of Schubert polynomials through ice . Adv. Math. 398(2022), Paper No. 108228. 29 pp.10.1016/j.aim.2022.108228CrossRefGoogle Scholar
Hodges, R. and Yong, A., Coxeter combinatorics and spherical Schubert geometry . J. Lie Theory 32(2022), 447–474.Google Scholar
Hodges, R. and Yong, A., Multiplicity-free key polynomials . Ann. Comb. 27(2023), 387–411.10.1007/s00026-022-00574-7CrossRefGoogle Scholar
Klein, P., Diagonal degenerations of matrix Schubert varieties . Algebraic Combin. 6(2023), 1073–1094.10.5802/alco.296CrossRefGoogle Scholar
Knutson, A., Lam, T., and Speyer, D. E., Positroid varieties: Juggling and geometry . Compositio Math. 149(2013), 1710–1752.10.1112/S0010437X13007240CrossRefGoogle Scholar
Kohnert, A., Weintrauben, Polynome, Tableaux . Bayreuth Math. Schrift. 38(1990), 1–97.Google Scholar
Kraskiewicz, W. and Pragacz, P., Foncteurs de Schubert . C. R. Acad. Sci. Paris Sér. I Math. 304(1987), 209–211.Google Scholar
Kraśkiewicz, W. and Pragacz, P., Schubert functors and Schubert polynomials . Eur. J. Combin. 25(2004), 1327–1344.10.1016/j.ejc.2003.09.016CrossRefGoogle Scholar
Knutson, A., Frobenius splitting and Möbius inversion. Preprint, 2009. arXiv:0902.1930v1.Google Scholar
Knutson, A. and Miller, E., Gröbner geometry of Schubert polynomials . Ann. Math. 161(2005), 1245–1318.10.4007/annals.2005.161.1245CrossRefGoogle Scholar
Magyar, P., Schubert polynomials and Bott–Samelson varieties . Comment. Math. Helv. 73(1998), 603–636.10.1007/s000140050071CrossRefGoogle Scholar
Mészáros, K., Setiabrata, L., and St Dizier, A., An orthodontia formula for Grothendieck polynomials . Trans. Amer. Math. Soc. 375(2022), 1281–1303.Google Scholar
Mészáros, K., Setiabrata, L., and St Dizier, A., On the support of Grothendieck polynomials . Ann. Comb. 29(2025), no. 2, 541–562.10.1007/s00026-024-00712-3CrossRefGoogle Scholar
Mészáros, K., Dizier, A. St., and Tanjaya, A., Principal specialization of dual characters of flagged Weyl modules . Electron. J. Combin. 28(2021), Paper No. 4.17. 12 pp.10.37236/10460CrossRefGoogle Scholar
Mészáros, K. and Tanjaya, A., Inclusion-exclusion on Schubert polynomials . Algebr. Comb. 5(2022), 209–226.Google Scholar
Pechenik, O. and Satriano, M., Proof of a conjectured Möbius inversion formula for Grothendieck polynomials . Selecta Math. (N.S.) 30(2024), Paper No. 83. 8 pp.10.1007/s00029-024-00973-zCrossRefGoogle Scholar
Peng, S. C. Y., Lin, Z., and Sun, S. C. C., Upper bounds of dual flagged Weyl characters . Adv. Appl. Math. 160(2024), 102752, 12pp.10.1016/j.aam.2024.102752CrossRefGoogle Scholar
Postnikov, A., Total positivity, Grassmannians, and networks. Preprint, 2006. http://arxiv.org/abs/math/0609764.Google Scholar