Hostname: page-component-7b9c58cd5d-wdhn8 Total loading time: 0 Render date: 2025-03-14T08:25:09.320Z Has data issue: false hasContentIssue false
  • English
  • Français

Double theta polynomials and equivariant Giambelli formulas

Published online by Cambridge University Press:  18 December 2015

HARRY TAMVAKIS  and
ELIZABETH WILSON
HARRY TAMVAKIS
Affiliation:
University of Maryland, Department of Mathematics, 1301 Mathematics Building, College Park, MD 20742, U.S.A. e-mail: harryt@math.umd.edu; bethmcl@math.umd.edu
ELIZABETH WILSON
Affiliation:
University of Maryland, Department of Mathematics, 1301 Mathematics Building, College Park, MD 20742, U.S.A. e-mail: harryt@math.umd.edu; bethmcl@math.umd.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We use Young's raising operators to introduce and study double theta polynomials, which specialize to both the theta polynomials of Buch, Kresch, and Tamvakis, and to double (or factorial) Schur S-polynomials and Q-polynomials. These double theta polynomials give Giambelli formulas which represent the equivariant Schubert classes in the torus-equivariant cohomology ring of symplectic Grassmannians, and we employ them to obtain a new presentation of this ring in terms of intrinsic generators and relations.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 160 , Issue 2 , March 2016 , pp. 353 - 377
Copyright
Copyright © Cambridge Philosophical Society 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[AF]Anderson, D. and Fulton, W. Chern class formulas for classical-type degeneracy loci. arXiv:1504.03615.Google Scholar
[BL]Biedenharn, L. C. and Louck, J. D.A new class of symmetric polynomials defined in terms of tableaux. Adv. Appl. Math. 10 (1989), 396–438.CrossRefGoogle Scholar
[BH]Billey, S. and Haiman, M.Schubert polynomials for the classical groups. J. Amer. Math. Soc. 8 (1995), 443–482.CrossRefGoogle Scholar
[BJS]Billey, S., Jockusch, W. and Stanley, R. P.Some combinatorial properties of Schubert polynomials. J. Algebraic Combin. 2 (1993), 345–374.CrossRefGoogle Scholar
[BKT1]Buch, A. S., Kresch, A. and Tamvakis, H.Quantum Pieri rules for isotropic Grassmannians. Invent. Math. 178 (2009), 345–405.CrossRefGoogle Scholar
[BKT2]Buch, A. S., Kresch, A. and Tamvakis, H. A Giambelli formula for isotropic Grassmannians. Selecta Math. (N.S.), to appear.Google Scholar
[BKT3]Buch, A. S., Kresch, A. and Tamvakis, H.Quantum Giambelli formulas for isotropic Grassmannians. Math. Ann. 354 (2012), 801–812.CrossRefGoogle Scholar
[BKT4]Buch, A. S., Kresch, A. and Tamvakis, H.A Giambelli formula for even orthogonal Grassmannians. J. reine angew. Math. 708 (2015), 17–48.CrossRefGoogle Scholar
[F]Fulton, W.Determinantal formulas for orthogonal and symplectic degeneracy loci. J. Differential Geom. 43 (1996), 276–290.CrossRefGoogle Scholar
[G]Giambelli, G. Z.Risoluzione del problema degli spazi secanti. Mem. R. Accad. Sci. Torino (2) 52 (1902), 171–211.Google Scholar
[Gr]Graham, W.The class of the diagonal in flag bundles. J. Differential Geom. 45 (1997), 471–487.CrossRefGoogle Scholar
[HIMN]Hudson, T., Ikeda, T., Matsumura, T. and Naruse, H. Determinantal and Pfaffian formulas of K-theoretic Schubert calculus, arXiv:1504.02828.Google Scholar
[I]Ikeda, T.Schubert classes in the equivariant cohomology of the Lagrangian Grassmannian. Adv. Math. 215 (2007), 1–23.CrossRefGoogle Scholar
[IM]Ikeda, T. and Matsumura, T.Pfaffian sum formula for the symplectic Grassmannian. Math. Z. 280 (2015), 269–306.CrossRefGoogle Scholar
[IMN1]Ikeda, T., Mihalcea, L. C. and Naruse, H.Double Schubert polynomials for the classical groups. Adv. Math. 226 (2011), 840–886.CrossRefGoogle Scholar
[IMN2]Ikeda, T., Mihalcea, L. C. and Naruse, H. Factorial P- and Q-Schur functions represent equivariant quantum Schubert classes, arXiv:1402.0892.Google Scholar
[IN]Ikeda, T. and Naruse, H.Excited Young diagrams and equivariant Schubert calculus. Trans. Amer. Math. Soc. 361 (2009), 5193–5221.CrossRefGoogle Scholar
[Iv]Ivanov, V. N.Interpolation analogues of Schur Q-functions, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 307 (2004), Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 10, 99–119, 281–282; translation in J. Math. Sci. (N. Y.) 131 (2005), 5495–5507.Google Scholar
[Ka]Kazarian, M. On Lagrange and symmetric degeneracy loci, preprint, Arnold Seminar (2000); available at http://www.newton.ac.uk/preprints/NI00028.pdf.Google Scholar
[KL]Kempf, G. and Laksov, D.The determinantal formula of Schubert calculus. Acta Math. 132 (1974), 153–162.CrossRefGoogle Scholar
[LS]Lascoux, A. and Schützenberger, M.-P.Polynômes de Schubert. C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), 447–450.Google Scholar
[M]Macdonald, I. G.Symmetric functions and Hall polynomials, Second edition, (The Clarendon Press, Oxford University Press, New York, 1995).CrossRefGoogle Scholar
[Mi1]Mihalcea, L. C.On equivariant quantum cohomology of homogeneous spaces: Chevalley formulae and algorithms. Duke Math. J. 140 (2007), 321–350.CrossRefGoogle Scholar
[Mi2]Mihalcea, L. C.Giambelli formulae for the equivariant quantum cohomology of the Grassmannian. Trans. Amer. Math. Soc. 360 (2008), 2285–2301.CrossRefGoogle Scholar
[P]Pragacz, P.Algebro-geometric applications of Schur S- and Q-polynomials, Séminare d'Algèbre Dubreil-Malliavin 1989-1990. Lecture Notes in Math. 1478, (Springer-Verlag, Berlin, 1991), 130–191.Google Scholar
[St]Stembridge, J. R.Some combinatorial aspects of reduced words in finite Coxeter groups. Trans. Amer. Math. Soc. 349 (1997), 1285–1332.CrossRefGoogle Scholar
[T1]Tamvakis, H.Giambelli, Pieri, and tableau formulas via raising operators. J. reine angew. Math. 652 (2011), 207–244.Google Scholar
[T2]Tamvakis, H.A Giambelli formula for classical G/P spaces. J. Algebraic Geom. 23 (2014), 245–278.CrossRefGoogle Scholar
[T3]Tamvakis, H. Giambelli and degeneracy locus formulas for classical G/P spaces. Mosc. Math. J., to appear.Google Scholar
[T4]Tamvakis, H. Double eta polynomials and equivariant Giambelli formulas, arXiv:1506.04441.Google Scholar
[W]Wilson, E. Equivariant Giambelli formulae for Grassmannians, PhD. thesis. University of Maryland (2010).Google Scholar
[Y]Young, A.On quantitative substitutional analysis VI. Proc. Lond. Math. Soc. (2) 34 (1932), 196–230.CrossRefGoogle Scholar