Skip to main content
SpringerLink
Log in

Polynomials for \(\mathrm{GL}_p\times \mathrm{GL}_q\) orbit closures in the flag variety

  • Published:
Selecta Mathematica Aims and scope Submit manuscript

Abstract

The subgroup \(K=\mathrm{GL}_p \times \mathrm{GL}_q\) of \(\mathrm{GL}_{p+q}\) acts on the (complex) flag variety \(\mathrm{GL}_{p+q}/B\) with finitely many orbits. We introduce a family of polynomials specializing representatives for cohomology classes of the orbit closures in the Borel model. We define and study \(K\)-orbit determinantal ideals to support the geometric naturality of these representatives. Using a modification of these ideals, we describe an analogy between two local singularity measures: the \(H\)-polynomials and the Kazhdan–Lusztig–Vogan polynomials.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Anderson, D., Griffeth, S., Miller, E.: Positivity and Kleiman transversality in equivariant K-theory of homogeneous spaces. J. Eur. Math. Soc. 13, 57–84 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bergeron, N., Billey, S.: RC-graphs and Schubert poylnomials. Exp. Math. 2, 257–269 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bernstein, I.N., Gelfand, I.M., Gelfand, S.I.: Schubert cells, and the cohomology of the spaces \(G/P\). Uspehi Mat. Nauk 28, 3(171)–26 (1973)

    Google Scholar 

  4. Borel, A.: Sur la cohomologie des espaces fibrés principaux et des espaces homogeènes de groupes de Lie compacts. Ann. Math. 57, 115–207 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brion, M.: On orbit closures of spherical subgroups in flag varieties. Comment. Math. Helv. 76(2), 263–299 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Springer, Berlin (1995)

    MATH  Google Scholar 

  7. Fomin, S., Gelfand, S., Postnikov, A.: Quantum Schubert polynomials. J. Am. Math. Soc. 10(3), 565–596 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fomin, S., Kirillov, A.: The Yang-Baxter equation, symmetric functions, and Schubert polynomials. Discrete Math. 153 123–143 (1996), Proceedings of Fifth Conference Formal Power Series and Algebraic Combinatorics (Florence, 1993)

  9. Fomin, S., Kirillov, A.: Grothendieck polynomials and the Yang–Baxter equation. In: Proceedings of 6th International Conference on Formal Power Series and Algebraic Combinatorics, DIMACS, pp. 183–190 (1994)

  10. Fulton, W.: With Applications to Representation Theory and Geometry, London Mathematical Society Student Texts, 35, Cambridge University Press, Cambridge (1997)

    Google Scholar 

  11. Fulton, W., Lascoux, A.: A Pieri formula in the Grothendieck ring of the flag bundle. Duke Math. J. 76(3), 711–729 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  12. Harada, M., Landweber, G.: Surjectivity for Hamiltonian \(G\)-spaces in \(K\)-theory. Trans. AMS 359, 6001–6025 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Insko, E., Yong, A.: Patch ideals and Peterson varieties. Transf. Groups 17(4), 1011–1036 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Knutson, A.: Frobenius splitting and Möbius inversion, preprint, arXiv:1209.4146 (2009)

  15. Knutson, A., Miller, E.: Gröbner geometry of Schubert polynomials. Ann. Math. 161, 1245–1318 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Knutson, A., Miller, E., Yong, A.: Gröbner geometry of vertex decompositions and of flagged tableaux. J. Reine Angew. Math. 630, 1–31 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Knutson, A., Woo, A., Yong, A.: Singularities of Richardson varieties, Math. Res. Lett. to appear arXiv:1209.4146 (2013)

  18. Kostant, B., Kumar, S.: \(T\)-equivariant \(K\)-theory of generalized flag varieties. J. Differ. Geom. 32(2), 549–603 (1990)

    MathSciNet  MATH  Google Scholar 

  19. Lascoux, A., Schützenberger, M.-P.: Polynômes de Schubert. C. R. Acad. Sci. Paris Sér. I Math. 295, 629–633 (1982)

    MATH  Google Scholar 

  20. Lenart, C., Sottile, F.: Skew Schubert polynomials. Proc. Am. Math. Soc. 131, 3319–3328 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Li, L., Yong, A.: Kazhdan–Lusztig polynomials and drift configurations. Algeb. Number Theory J. 5(5), 595–626 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lusztig, G., Vogan, D.: Singularities of closures of \(K\)-orbits on flag manifolds. Invent. Math. 71(2), 365–379 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  23. Manivel, L.: Symmetric Functions, Schubert Polynomials and Degeneracy Loci. American Mathematical Society, Providence, RI (2001)

    MATH  Google Scholar 

  24. Matsuki, T.: The orbits of affine symmetric spaces under the action of minimal parabolic subgroups. J. Math. Soc. Jpn. 31(2), 331–357 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  25. Matsuki, T., Oshima, T.: Embeddings of discrete series into principal series. In: The Orbit Method in Representation Theory (Copenhagen, 1988), Volume 82 of Progr. Math., pp. 147–175. Birkhäuser, Boston, MA (1990)

  26. McGovern, W.M.: Closures of \(K\)-orbits in the flag variety for \(U(p, q)\). J. Algeb. 322(8), 2709–2712 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. McGovern, W. M.: Upper semicontinuity of KLV polynomials for certain blocks of Harish–Chandra modules. Preprint. arXiv:1311.0911 (2013)

  28. McGovern, W.M., Trapa, P.: Pattern avoidance and smoothness of closures for orbits of a symmetric subgroup in the flag variety. J. Algeb. 322(8), 2713–2730 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Miller, E., Sturmfels, B.: Combinatorial Commutative Algebra, Graduate Texts in Mathematics, vol. 227. Springer, New York (2004)

    Google Scholar 

  30. Richardson, R.W.: Intersections of double cosets in algebraic groups. Indag. Math. (N.S.) 3(1), 69–77 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  31. Richardson, R.W., Springer, T.A.: The Bruhat order on symmetric varieties. Geom. Dedic. 35, 389–436 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  32. Springer, T.A.: Some results on algebraic groups with involutions. Algebraic groups and related topics, Vol. 6 of Adv. Stud. Pure Math, pp. 525–543

  33. Vogan, D.: Irreducible characters of semisimple Lie groups. III. Proof of Kazhdan–Lusztig conjecture in the integral case. Invent. Math. 71(2), 381–417 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wyser, B.: \(K\)-orbit closures on \(G/B\) as universal degeneracy loci for flagged vector bundles splitting as direct sums. Preprint. arXiv:1301.1713 (2013)

  35. Wyser, B.: \(K\)-orbit closures on \(G/B\) as universal degeneracy loci for flagged vector bundles with symmetric or skew-symmetric bilinear form. Transf. Groups 18(2), 557–594 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wyser, B.: Schubert calculus of Richardson varieties stable under spherical Levi subgroups. J. Algeb. Comb. 38(4), 829–850 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wyser, B., Yong, A.: Polynomials for symmetric orbit closures in the flag variety. Preprint. arXiv:1310.7271 (2013)

  38. Yamamoto, A.: Orbits in the flag variety and images of the moment map for classical groups. I. Represent. Theory 1, 329–404 (1997). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

We wish to thank Bill Graham, Allen Knutson, William McGovern, Oliver Pechenik, Hal Schenck, Peter Trapa, Hugh Thomas, and Alexander Woo for helpful correspondence. We also thank the anonymous referee for his/her useful suggestions. A.Y. was supported by NSF Grants. This text was completed while A.Y. was a Helen Corley Petit scholar at UIUC.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

    Benjamin J. Wyser & Alexander Yong

Authors
  1. Benjamin J. Wyser
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexander Yong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Yong.

Appendix

Appendix

Below we give the polynomials \(\Upsilon _{\gamma }(X;Y)\) for all \(\gamma \in \mathtt{Clans}_{2,2}\).

figure k

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wyser, B.J., Yong, A. Polynomials for \(\mathrm{GL}_p\times \mathrm{GL}_q\) orbit closures in the flag variety. Sel. Math. New Ser. 20, 1083–1110 (2014). https://doi.org/10.1007/s00029-014-0152-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00029-014-0152-z

Keywords

Mathematical Subject Classification (2010)

Search

Navigation