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A Giambelli formula for isotropic Grassmannians

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Abstract

Let X be a symplectic or odd orthogonal Grassmannian. We prove a Giambelli formula which expresses an arbitrary Schubert class in \({{\mathrm{\mathrm {H}}}}^*(X,{\mathbb Z})\) as a polynomial in certain special Schubert classes. This polynomial, which we call a theta polynomial, is defined using raising operators, and we study its image in the ring of Billey–Haiman Schubert polynomials.

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Notes

  1. As is customary, we slightly abuse the notation and consider that the raising operator R acts on the index \(\alpha \), and not on the monomial \(m_\alpha \) itself.

  2. We use the term ‘theta polynomial’ to denote both the Giambelli polynomial in Theorem 1 and its image in \(\Gamma ^{(k)}\); see Definition 5.3.

References

  1. Bergeron, N., Sottile, F.: A Pieri-type formula for isotropic flag manifolds. Trans. Am. Math. Soc. 354, 4815–4829 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Billey, S., Haiman, M.: Schubert polynomials for the classical groups. J. Am. Math. Soc. 8, 443–482 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. Buch, A.S., Kresch, A., Tamvakis, H.: Quantum Pieri rules for isotropic Grassmannians. Invent. Math. 178, 345–405 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Buch, A.S., Kresch, A., Tamvakis, H.: Quantum Giambelli formulas for isotropic Grassmannians. Math. Ann. 354, 801–812 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Buch, A.S., Kresch, A., Tamvakis, H.: A Giambelli formula for even orthogonal Grassmannians. J. Reine Angew. Math. 708, 17–48 (2015)

    MathSciNet  MATH  Google Scholar 

  6. Cauchy, A.L.: Mémoire sur les fonctions qui ne peuvent obtenir que deux valeurs égales et de signes contraires par suite des transpositions opérés entre les variables qu’elles renferment. J. École Polyt. 10, 29–112 (1815); Oeuvres, ser. 2, vol. 1, 91–169

  7. Fomin, S., Kirillov, A.N.: Combinatorial \(B_n\)-analogs of Schubert polynomials. Trans. Am. Math. Soc. 348, 3591–3620 (1996)

    Article  MATH  Google Scholar 

  8. Giambelli, G.Z.: Risoluzione del problema degli spazi secanti. Mem. R. Accad. Sci. Torino 2(52), 171–211 (1902)

    MATH  Google Scholar 

  9. Hoffman, P.N., Humphreys, J.F.: Projective Representations of the Symmetric Groups; \(Q\)-Functions and Shifted Tableaux. Oxford University Press, New York (1992)

    MATH  Google Scholar 

  10. Jacobi, C.G.J.: De functionibus alternantibus earumque divisione per productum e differentiis elementorum conflatum. J. Reine Angew. Math. 22, 360–371 (1841); Reprinted in Gesammelte Werke 3, 439–452, Chelsea, New York (1969)

  11. Kraśkiewicz, W.: Reduced decompositions in hyperoctahedral groups. C. R. Acad. Sci. Paris Sér. I Math. 309, 903–907 (1989)

    MathSciNet  MATH  Google Scholar 

  12. Lam, T.K.: \(B_n\) Stanley symmetric functions. In: Proceedings of the 6th Conference on Formal Power Series and Algebraic Combinatorics (New Brunswick, NJ, 1994), Discrete Math. 157, 241–270 (1996)

  13. Lascoux, A., Schützenberger, M.P.: Polynômes de Schubert. C. R. Acad. Sci. Paris Sér. I Math. 294, 447–450 (1982)

    MathSciNet  MATH  Google Scholar 

  14. Macdonald, I.: Symmetric Functions and Hall Polynomials, 2nd edn. Clarendon Press, Oxford (1995)

    MATH  Google Scholar 

  15. Pragacz, P.: Algebro-geometric applications of Schur \(S\)- and \(Q\)-polynomials, Séminare d’Algèbre Dubreil-Malliavin 1989–1990. Springer Lecture Notes in Mathematics, vol. 1478, pp. 130–191 (1991)

  16. Pragacz, P., Ratajski, J.: A Pieri-type theorem for Lagrangian and odd orthogonal Grassmannians. J. Reine Angew. Math. 476, 143–189 (1996)

    MathSciNet  MATH  Google Scholar 

  17. Schur, I.: Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen. J. Reine Angew. Math. 139, 155–250 (1911)

    MathSciNet  MATH  Google Scholar 

  18. Tamvakis, H.: Quantum cohomology of isotropic Grassmannians, Geometric Methods in Algebra and Number Theory, pp. 311–338, Progress in Math. 235, Birkhäuser (2005)

  19. Tamvakis, H.: Giambelli, Pieri, and tableau formulas via raising operators. J. Reine Angew. Math. 652, 207–244 (2011)

    MathSciNet  MATH  Google Scholar 

  20. Worley, D.R.: A Theory of Shifted Young Tableaux. Ph.D. thesis, MIT (1984)

  21. Young, A.: On quantitative substitutional analysis VI. Proc. Lond. Math. Soc. 34(2), 196–230 (1932)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ, 08854, USA

    Anders Skovsted Buch

  2. Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057, Zurich, Switzerland

    Andrew Kresch

  3. Department of Mathematics, University of Maryland, 1301 Mathematics Building, College Park, MD, 20742, USA

    Harry Tamvakis

Authors
  1. Anders Skovsted Buch
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  2. Andrew Kresch
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  3. Harry Tamvakis
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Corresponding author

Correspondence to Harry Tamvakis.

Additional information

The authors were supported in part by NSF Grants DMS-0603822 and DMS-0906148 (Buch), the Swiss National Science Foundation (Kresch), and NSF Grants DMS-0639033 and DMS-0901341 (Tamvakis).

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Buch, A.S., Kresch, A. & Tamvakis, H. A Giambelli formula for isotropic Grassmannians. Sel. Math. New Ser. 23, 869–914 (2017). https://doi.org/10.1007/s00029-016-0250-1

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