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Variations on a Theme of Schubert Calculus

  • Maria GillespieEmail author
Chapter
Part of the Association for Women in Mathematics Series book series (AWMS, volume 16)

Abstract

In this tutorial, we provide an overview of many of the established combinatorial and algebraic tools of Schubert calculus, the modern area of enumerative geometry that encapsulates a wide variety of topics involving intersections of linear spaces. It is intended as a guide for readers with a combinatorial bent to understand and appreciate the geometric and topological aspects of Schubert calculus, and conversely for geometric-minded readers to gain familiarity with the relevant combinatorial tools in this area. We lead the reader through a tour of three variations on a theme: Grassmannians, flag varieties, and orthogonal Grassmannians. The orthogonal Grassmannian, unlike the ordinary Grassmannian and the flag variety, has not yet been addressed very often in textbooks, so this presentation may be helpful as an introduction to type B Schubert calculus. This work is adapted from the author’s lecture notes for a graduate workshop during the Equivariant Combinatorics Workshop at the Center for Mathematics Research, Montreal, June 12–16, 2017.

Notes

Acknowledgements

The author thanks Jennifer Morse, François Bergeron, Franco Saliola, and Luc Lapointe for the invitation to teach a graduate workshop on Schubert calculus at the Center for Mathematics Research in Montreal, for which these extended notes were written. Thanks also to Sara Billey and the anonymous reviewer for their extensive feedback. Thanks to Helene Barcelo, Sean Griffin, Philippe Nadeau, Alex Woo, Jake Levinson, and Guanyu Li for further suggestions and comments. Finally, thanks to all of the participants at the graduate workshop for their comments, questions, and corrections that greatly improved this exposition.

References

  1. 1.
    H. Abe, S. Billey, Consequences of the Lakshmibai-Sandhya Theorem: The ubiquity of permutation patterns in Schubert calculus and related geometry (2014), arxiv:1403.4345
  2. 2.
    D. Anderson, Introduction to equivariant cohomology in algebraic geometry, Notes on lectures by W. Fulton at IMPAGNA summer school (2010), arXiv:1112.1421
  3. 3.
    D. Anderson, L. Chen, Equivariant quantum Schubert polynomials. Adv. Math. 254, 300–330 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A.B. Goncharov, A. Postnikov, J. Trnka, Grassmannian Geometry of Scattering Amplitudes (Cambridge University Press, Cambridge, 2016)CrossRefGoogle Scholar
  5. 5.
    F. Bergeron, Algebraic Combinatorics and Coinvariant Spaces (CRC Press, Baco Raton, 2009)CrossRefGoogle Scholar
  6. 6.
    I.N. Bernstein, I.M. Gelfand, S.I. Gelfand, Schubert cells and cohomology of the spaces \(G/P\). Russ. Math. Surv. 28(3), 1–26 (1973)MathSciNetCrossRefGoogle Scholar
  7. 7.
    A. Bertiger, E. Milićević, K. Taipale, Equivariant quantum cohomology of the Grassmannian via the rim hook rule (2014), arxiv:1403.6218
  8. 8.
    S. Billey, M. Haiman, Schubert polynomials for the classical groups. J. Amer. Math. Soc. 8(2) (1995)MathSciNetCrossRefGoogle Scholar
  9. 9.
    S. Billey, V. Lakshmibai, Singular Loci of Schubert Varieties (Springer, Berlin, 2000)CrossRefGoogle Scholar
  10. 10.
    A. Bjorner, F. Brenti, Combinatorics of Coxeter Groups (Springer, Berlin, 2005)zbMATHGoogle Scholar
  11. 11.
    J. Bourjaily, H. Thomas, What is the Amplituhedron? Not. AMS 65, 167–169 (2018)MathSciNetzbMATHGoogle Scholar
  12. 12.
    A. Buch, A Littlewood–Richardson rule for the \(K\)-theory of Grassmannians. Acta Math. 189(1), 37–78 (2002)MathSciNetCrossRefGoogle Scholar
  13. 13.
    A. Buch, A. Kresch, K. Purbhoo, H. Tamvakis, The puzzle conjecture for the cohomology of two-step flag manifolds. J. Alg. Comb. 44(4), 973–1007 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    L. Chen, Quantum cohomology of flag manifolds. Adv. Math. 174(1), 1–34 (2003)MathSciNetCrossRefGoogle Scholar
  15. 15.
    I. Coskun, A Littlewood–Richardson rule for partial flag varieties, preprint (under revision), http://www.homepages.math.uic.edu/~coskun/newpartial.pdf
  16. 16.
    I. Coskun, A Littlewood–Richardson rule for two-step flag varieties. Invent. Math. 176, 325 (2009)MathSciNetCrossRefGoogle Scholar
  17. 17.
    D. Cox, J. Little, D. O’Shea, Ideals, Varieties, and Algorithms, vol. 2 (Springer, Berlin, 1991)zbMATHGoogle Scholar
  18. 18.
    J.S. Frame, G.de B. Robinson, R.M. Thrall, The hook graphs of the symmetric group. Can. J. Math. 6, 316–325 (1954)MathSciNetCrossRefGoogle Scholar
  19. 19.
    S. Fomin, S. Gelfand, Quantum Schubert polynomials. J. Amer. Math. Soc. 10, 565–596 (1997)MathSciNetCrossRefGoogle Scholar
  20. 20.
    S. Fomin, A. Kirillov, Combinatorial \(B_n\)-analogues of Schubert polynomials. Trans. Am. Math. Soc. 348(9) (1996)Google Scholar
  21. 21.
    W. Fulton, Intersection Theory, vol. 2 (Springer, Berlin, 1998)CrossRefGoogle Scholar
  22. 22.
    W. Fulton, Young Tableaux, with Applications to Representation Theory and Geometry (Cambridge University Press, Cambridge, 1997)Google Scholar
  23. 23.
    A.M. Garsia, C. Procesi, On certain graded \(S_n\)-modules and the \(q\)-Kostka polynomials. Adv. Math. 94(1), 82–138 (1992)MathSciNetCrossRefGoogle Scholar
  24. 24.
    V. Gasharov, V. Reiner, Cohomology of smooth Schubert varieties in partial flag manifolds. J. Lond. Math. Soc. 66(3), 550–562 (2002)MathSciNetCrossRefGoogle Scholar
  25. 25.
    M. Gillespie, J. Levinson, K. Purbhoo, A crystal-like structure on shifted tableaux (2017), arxiv:1706.09969
  26. 26.
    R. Green, Combinatorics of Minuscule Representations (Cambridge Tracts in Mathematics 199) (Cambridge University Press, Cambridge, 2013)Google Scholar
  27. 27.
    A. Hatcher, Algebraic Topology (Cambridge University Press, Cambridge, 2001)zbMATHGoogle Scholar
  28. 28.
    S. Kleiman, D. Laskov, Schubert calculus. Am. Math. Mon. 79(10), 1061–1082 (1972)MathSciNetCrossRefGoogle Scholar
  29. 29.
    A. Knutson, T. Tao, C. Woodward, The honeycomb model of \(GL_n({\mathbb{C}})\) tensor products II: puzzles determine facets of the Littlewood–Richardson cone. J. Am. Math. Soc. 17, 19–48 (2004)CrossRefGoogle Scholar
  30. 30.
    A. Knutson, P. Zinn-Justin, Schubert, puzzles and integrability I: invariant trilinear forms (2017), arxiv:1706.10019v4
  31. 31.
    E.R. Kolchin, Algebraic matric groups and the Picard–Vessiot theory of homogeneous linear ordinary differential equations. Ann. Math. Second Ser. 49, 1–42 (1948)MathSciNetCrossRefGoogle Scholar
  32. 32.
    M. Kontsevich, Yu. Manin, Gromov–Witten classes, quantum cohomology, and enumerative geometry. Commun. Math. Phys. 164, 525–562 (1994)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Kac-Moody Groups, Their Flag Varieties and Representation Theory, vol. 204, Progress in Mathematics (Birkhäuser, Basel, 2002)Google Scholar
  34. 34.
    T. Lam, L. Lapointe, J. Morse, A. Schilling, M. Shimozono, M. Zabrocki, \(k\)-Schur Functions and Affine Schubert Calculus (Springer, Berlin, 2014)zbMATHGoogle Scholar
  35. 35.
    V. Lakshmibai, B. Sandhya, Criterion for smoothness of Schubert varieties in \(SL(n)/B\). Proc. Indian Acad. Sci. Math. Sci. 100, 45 (1990)Google Scholar
  36. 36.
    A. Lascoux, M.-P. Schützenberger, Polynômes de Schubert, C. R. Acad. Sci. Paris Sér. I Math. 294(13), 447–450 (1982)Google Scholar
  37. 37.
    S.J. Lee, Combinatorial description of the cohomology of the affine flag variety. Trans. Am. Math. Soc. arXiv:1506.02390 (to appear)
  38. 38.
    I. Macdonald, Symmetric Functions and Hall Polynomials (Oxford University Press, Oxford, 1979)zbMATHGoogle Scholar
  39. 39.
    L. Manivel, Symmetric Functions, Schubert Polynomials, and Degeneracy Loci, American Mathematical Society (2001)Google Scholar
  40. 40.
    E. Mukhin, V. Tarasov, A. Varchenko, Schubert calculus and representations of the general linear group. J. Am. Math. Soc. 22(4), 909–940 (2009)MathSciNetCrossRefGoogle Scholar
  41. 41.
    O. Pechenik, A. Yong, Equivariant K-theory of Grassmannians. Forum Math. Pi 5, 1–128 (2017)MathSciNetCrossRefGoogle Scholar
  42. 42.
    O. Pechenik, A. Yong, Equivariant K-theory of Grassmannians II: the Knutson–Vakil conjecture. Compos. Math. 153, 667–677 (2017)MathSciNetCrossRefGoogle Scholar
  43. 43.
    A. Postnikov, D. Speyer, L. Williams, Matching polytopes, toric geometry, and the totally non-negative Grassmannian. J. Alg. Comb. 30, 173–191 (2009)MathSciNetCrossRefGoogle Scholar
  44. 44.
    P. Pragacz, Algebro-geometric applications of Schur \(S\)- and \(Q\)-polynomials, in Topics in Invariant Theory, vol. 1478, Springer Lecture Notes in Mathematics, ed. by M.P. Malliavin (Springer, Berlin, 1991), pp. 130–191CrossRefGoogle Scholar
  45. 45.
    F. Ronga, Schubert calculus according to Schubert (2006), arxiv:0608784
  46. 46.
    Y. Ruan, G. Tian, Mathematical theory of quantum cohomology. J. Differ. Geom. 42(2), 259–367 (1995)MathSciNetCrossRefGoogle Scholar
  47. 47.
    K. Ryan, On Schubert varieties in the flag manifold of \(SL(n,\mathbb{C})\). Math. Ann. 276, 205–224 (1987)MathSciNetCrossRefGoogle Scholar
  48. 48.
    The Symmetric Group, vol. 2 (Springer, New York, 2001)Google Scholar
  49. 49.
    H. Schubert, Kalkül der abzählende Geometrie (Teubner Verlag, Leipzig, 1789.)Google Scholar
  50. 50.
    F. Sottile, Frontiers of reality in Schubert calculus. Bull. AMS 47(1), 31–71 (2010)MathSciNetCrossRefGoogle Scholar
  51. 51.
    R. Stanley, Enumerative Combinatorics, vol. 2 (Cambridge University Press, Cambridge, 1999)CrossRefGoogle Scholar
  52. 52.
    Adv. Math. Shifted tableaux and the projective representations of the symmetric group. 74(1), 87–134 (1989)Google Scholar
  53. 53.
    T. Tajakka, Cohomology of the Grassmannian. Master’s thesis, Aalto University, 2015Google Scholar
  54. 54.
    H. Thomas, A. Yong, A combinatorial rule for (co)minuscule Schubert calculus. Adv. Math. 222(2), 596–620 (2009)MathSciNetCrossRefGoogle Scholar
  55. 55.
    J. Tymoczko, Decomposing Hessenberg varieties over classical groups. Ph.D. thesis, 2010, arxiv:0211226
  56. 56.
    R. Vakil, A geometric Littlewood–Richardson rule. Ann. Math. 164(2), 371–422 (2006)MathSciNetCrossRefGoogle Scholar
  57. 57.
    J.S. Wolper, A combinatorial approach to the singularities of Schubert varieties. Adv. Math. 76, 184–193 (1989)MathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) and the Association for Women in Mathematics 2019

Authors and Affiliations

  1. 1.University of CaliforniaDavisUSA

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