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K-Orbit Closures on G/B as Universal Degeneracy Loci for Flagged Vector Bundles with Symmetric or Skew-Symmetric Bilinear Form

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Abstract

We use equivariant localization and divided difference operators to determine formulas for the torus-equivariant fundamental cohomology classes of K-orbit closures on the flag variety G/B, where G = GL(n, \( \mathbb{C} \)), and where K is one of the symmetric subgroups O(n, \( \mathbb{C} \)) or Sp(n, \( \mathbb{C} \)). We realize these orbit closures as universal degeneracy loci for a vector bundle over a variety equipped with a single flag of subbundles and a nondegenerate symmetric or skew-symmetric bilinear form taking values in the trivial bundle. We describe how our equivariant formulas can be interpreted as giving formulas for the classes of such loci in terms of the Chern classes of the various bundles.

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References

  1. M. Brion, Equivariant cohomology and equivariant intersection theory, in: Representation Theories and Algebraic Geometry, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 514, Kluwer, 1997, pp. 1–37.

  2. M. Brion, Rational smoothness and fixed points of torus actions, Transform. Groups 4 (1999), no. 2&3, 127–156.

    Article  MathSciNet  MATH  Google Scholar 

  3. M. Brion, On orbit closures of spherical subgroups in flag varieties, Comm. Math. Helv. 76 (2001), no. 2, 263–299.

    Article  MathSciNet  MATH  Google Scholar 

  4. D. Edidin, W. Graham, Characteristic classes and quadric bundles, Duke Math. J. 78 (1995), no. 2, 277–299.

    Article  MathSciNet  MATH  Google Scholar 

  5. W. Fulton, Flags, Schubert polynomials, degeneracy loci, and determinantal formulas, Duke Math. J. 65 (1992), no. 3, 381–420.

    Article  MathSciNet  MATH  Google Scholar 

  6. W. Fulton, Schubert varieties in flag bundles for the classical groups, in: Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), Israel Math. Conf. Proc., Vol. 9, Bar-Ilan Univ., 1996, pp. 241–262.

  7. W. Fulton, Determinantal formulas for orthogonal and symplectic degeneracy loci, J. Diff. Geom. 43 (1996), no. 2, 276–290.

    MathSciNet  MATH  Google Scholar 

  8. W. Fulton, Intersection Theory, 2nd ed., Springer-Verlag, Berlin, 1998.

    Book  MATH  Google Scholar 

  9. W. Fulton, Young Tableaux, Cambridge University Press, Cambridge, 1997.

    MATH  Google Scholar 

  10. W. Graham, The class of the diagonal in flag bundles, J. Diff. Geom. 45 (1997), no. 3, 471–487.

    MATH  Google Scholar 

  11. J. Humphreys, Linear Algebraic Groups, Springer-Verlag, New York, 1975. Russian transl.: Дж. Хамфри, Линейные алгебраические группы, Нayкa, M., 1980.

    Book  MATH  Google Scholar 

  12. T. Matsuki, The orbits of affine symmetric spaces under the action of minimal parabolic subgroups, J. Math. Soc. Japan 31 (1979), no. 2, 331–357.

    Article  MathSciNet  MATH  Google Scholar 

  13. W. McGovern, P. Trapa, Pattern avoidance and smoothness of closures for orbits of a symmetric subgroup in the fag variety, J. Algebra 322 (2009), no. 8, 2713–2730.

    Article  MathSciNet  MATH  Google Scholar 

  14. J. Milnor, J. Stasheff, Characteristic Classes, Princeton University Press, Princeton, NJ, 1974. Russian transl.: Дж. Милнор, Дж. Cтaшeф, Характеристические классы, Мир, M., 1979.

    MATH  Google Scholar 

  15. R. W. Richardson, T. A. Springer, The Bruhat order on symmetric varieties, Geom. Dedicata 35 (1990), no. 1–3, 389–436.

    MathSciNet  MATH  Google Scholar 

  16. R. W. Richardson, T. A. Springer, Combinatorics and geometry of K-orbits on the flag manifold, in: Linear Algebraic Groups and their Representations (Los Angeles, CA, 1992), Contemp. Math., Vol. 153, Amer. Math. Soc., 1993, pp. 109–142.

  17. R. W. Richardson, T. A. Springer, Complements to: “The Bruhat order on symmetric varieties” [Geom. Dedicata 35 (1990), no. 1–3, 389–436], Geom. Dedicata 49 (1994), no. 2, 231–238.

    Article  MathSciNet  MATH  Google Scholar 

  18. T. A. Springer, Some results on algebraic groups with involutions, in: Algebraic Groups and Related Topics (Kyoto/Nagoya, 1983), Adv. Stud. Pure Math., Vol. 6, North-Holland, 1985, pp. 525–543.

  19. B. Wyser, Symmetric subgroup orbit closures on flag varieties: Their equivariant geometry, combinatorics, and connections with degeneracy loci, PhD thesis, University of Georgia, 2012.

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  1. Department of Mathematics, University of Illinois at Urbana−Champaign, 250 Altgeld Hall 1409 W. Green St., Urbana, IL, 61801, USA

    Benjamin J. Wyser

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  1. Benjamin J. Wyser
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Correspondence to Benjamin J. Wyser.

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Dedicated to my daughters, Avery and Carsyn

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Wyser, B.J. K-Orbit Closures on G/B as Universal Degeneracy Loci for Flagged Vector Bundles with Symmetric or Skew-Symmetric Bilinear Form. Transformation Groups 18, 557–594 (2013). https://doi.org/10.1007/s00031-013-9221-1

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  • DOI: https://doi.org/10.1007/s00031-013-9221-1

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