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Affine Grassmannians and Hessenberg Schubert Cells

  • Linda ChenEmail author
  • Julianna Tymoczko
Chapter
Part of the Association for Women in Mathematics Series book series (AWMS, volume 16)

Abstract

We give an overview of the linear algebra, geometry, and combinatorics of affine Grassmannians along the lines of Fulton’s Young Tableaux for classical Grassmannians. We discuss geometric and linear algebraic aspects of the decomposition of the affine Grassmannian into affine Schubert cells in terms of coset representatives and linear models. We describe (Grassmannian) Hessenberg Schubert cells and show that every affine Schubert cell can be realized as a Hessenberg Schubert cell in a complete flag variety and as a Grassmannian Hessenberg Schubert cell in a finite Grassmannian.

References

  1. 1.
    M. Atiyah, I. MacDonald, Introduction to Commutative Algebra (Addison-Wesley, Reading, 1969)zbMATHGoogle Scholar
  2. 2.
    H. Abe, M. Harada, T. Horiguchi, M. Masuda, The cohomology rings of regular nilpotent Hessenberg varieties in Lie type A, arXiv:1512.09072
  3. 3.
    Darius Bayegan, Megumi Harada, A Giambelli formula for the \(S^1\)-equivariant cohomology of type A Peterson varieties. Involve 5, 115–132 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Sara Billey, Steve Mitchell, Smooth and palindromic Schubert varieties in affine Grassmannians. J. Algebr. Comb. 31, 169–216 (2010)MathSciNetCrossRefGoogle Scholar
  5. 5.
    F. De Mari, C. Procesi, M. Shayman, Hessenberg varieties. Trans. Am. Math. Soc. 332, 529–534 (1992)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Elizabeth Drellich, Monk’s rule and Giambelli’s formula for Peterson varieties of all Lie types. J. Algebr. Comb. 41, 539–575 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Lucas Fresse, A unified approach on Springer fibers in the hook, two-row and two-column cases. Trans. Groups 15(2), 285–331 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Lucas Fresse, Anna Melnikov, On the singularity of the irreducible components of a Springer fiber in \(mathfrak{sl}_n\). Sel. Math. (N.S.) 16(3), 393–418 (2010)CrossRefGoogle Scholar
  9. 9.
    W. Fulton, Tableaux Young, with Applications to Representation Theory and Geometry, vol. 35, London Mathematical Society Student Texts (Cambridge University Press, Cambridge, 1997)Google Scholar
  10. 10.
    Y.C. Francis, Fung, On the topology of components of some Springer fibers and their relation to Kazhdan–Lusztig theory. Adv. Math. 178(2), 244–276 (2003)MathSciNetCrossRefGoogle Scholar
  11. 11.
    A. Garsia, C. Procesi, On certain graded \(S_n\)-modules and the \(q\)-Kostka polynomials. Adv. Math. 94, 82–138 (1992)MathSciNetCrossRefGoogle Scholar
  12. 12.
    M. Khovanov, Crossingless matchings and the cohomology of \((n, n)\) Springer varieties. Commun. Contemp. Math. 6(4), 561–577 (2004)MathSciNetCrossRefGoogle Scholar
  13. 13.
    B. Kostant, Flag manifold quantum cohomology, the Toda lattice, and the representation with highest weight \(\rho \). Sel. Math. (N.S.) 2, 43–91 (1996)Google Scholar
  14. 14.
    B. Kostant, S. Kumar, The nil Hecke ring and cohomology of \(G/P\) for a Kac-Moody group \(G\). Adv. Math. 62, 187–237 (1986)Google Scholar
  15. 15.
    T. Lam, Schubert polynomials for the affine Grassmannian. J. Am. Math. Soc 21, 259–281 (2008)MathSciNetCrossRefGoogle Scholar
  16. 16.
    T. Lam, L. Lapointe, J. Morse, M. Shimozono, Affine insertion and Pieri rules for the affine Grassmannian. Am. J. Math. 128(6), 1553–1586 (2006)Google Scholar
  17. 17.
    T. Lam, L. Lapointe, J. Morse, A. Schilling, M. Shimozono, M. Zabrocki, \(k\)-Schur functions and affine Schubert calculus, vol. 33, Fields Institute Monographs (2014)zbMATHGoogle Scholar
  18. 18.
    T. Lam, M. Shimozono, Quantum cohomology of \(G/P\) and homology of affine Grassmannian. Acta. Math. 204, 49–90 (2010)MathSciNetCrossRefGoogle Scholar
  19. 19.
    L. Lapointe, J. Morse, Quantum cohomology and the \(k\)-Schur basis. Trans. Am. Math. Soc. 360, 2021–2040 (2008)Google Scholar
  20. 20.
    D. Peterson, Quantum cohomology of \({G}/{P}\) (Lecture notes, Massachusetts Institute of Technology, Cambridge, 1997)Google Scholar
  21. 21.
    M. Precup, The Betti numbers of regular Hessenberg varieties are palindromic. Trans. Groups 23, 491–499 (2018)MathSciNetCrossRefGoogle Scholar
  22. 22.
    K. Rietsch, Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties. J. Am. Math. Soc. 16, 363–392 (2003)MathSciNetCrossRefGoogle Scholar
  23. 23.
    T. Springer, Trigonometric sums, Green functions of finite groups and representations of Weyl groups. Invent. Math. 36, 173–207 (1976)MathSciNetCrossRefGoogle Scholar
  24. 24.
    J. Tymoczko, Linear conditions imposed on flag varieties. Am. J. Math. 128, 1587–1604 (2006)MathSciNetCrossRefGoogle Scholar
  25. 25.
    J. Tymoczko, Paving Hessenberg varieties by affines. Sel. Math. (N.S.) 13, 353–367 (2007)MathSciNetCrossRefGoogle Scholar
  26. 26.
    J. Tymoczko, Permutation representations on Schubert varieties. Am. J. Math. 130(5), 1171–1194 (2008)MathSciNetCrossRefGoogle Scholar
  27. 27.
    X. Zhu, An introduction to affine Grassmannians and the geometric Satake equivalence, arXiv:1603.05593 [math.AG]

Copyright information

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

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsSwarthmore CollegeSwarthmoreUSA
  2. 2.Department of Mathematics and StatisticsSmith CollegeNorthamptonUSA

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