Intersections of Schubert varieties and other permutation array schemes

  • Sara Billey
  • Ravi Vakil
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 146)


Using a blend of combinatorics and geometry, we give an algorithm for algebraically finding all flags in any zero-dimensional intersection of Schubert varieties with respect to three transverse flags, and more generally, any number of flags. The number of flags in a triple intersection is also a structure constant for the cohomology ring of the flag manifold. Our algorithm is based on solving a limited number of determinantal equations for each intersection (far fewer than the naive approach in the case of triple intersections). These equations may be used to compute Galois and monodromy groups of intersections of Schubert varieties. We are able to limit the number of equations by using the permutation arrays of Eriksson and Linusson, and their permutation array varieties, introduced as generalizations of Schubert varieties. We show that there exists a unique permutation array corresponding to each realizable Schubert problem and give a simple recurrence to compute the corresponding rank table, giving in particular a simple criterion for a Littlewood-Richardson coefficient to be 0. We describe pathologies of Eriksson and Linusson’s permutation array varieties (failure of existence, irreducibility, equidimensionality, and reducedness of equations), and define the more natural permutation array schemes. In particular, we give several counterexamples to the Readability Conjecture based on classical projective geometry. Finally, we give examples where Galois/monodromy groups experimentally appear to be smaller than expected.

Key words

Schubert varieties permutation arrays Littlewood-Richardson coefficients 

AMS(MOS) subject classifications

Primary 14M17 Secondary 14M15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Ardila and Billey, 2006]
    F. ARDILA AND S. BiLLEY, Flag arrangements and triangulations of products of simplices, to appear in Advances in Math.Google Scholar
  2. [Cohen, 1981]
    S.D. COHEN, The distribution of Galois groups and Hubert’s irreducibility theorem, Proc. London Math. Soc. (3) 43 (1981), no. 2, 227–250.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [Coskun]
    I. COSKUN, A Littlewood-Richarson rule for the two-step flag varieties, preprint, 2004.Google Scholar
  4. [Coskun and Vakil]
    I. COSKUN AND R. VAKIL, Geometric positivity in the co-homology of homogeneous spaces and generalized Schubert calculus, arXiv:math.AG/0610538.Google Scholar
  5. [Coxeter and Greitzer, 1967]
    H.S.M. COXETER AND S.L. GREITZER, Geometry Revisited, Math. Ass. of Amer., New Haven, 1967.zbMATHGoogle Scholar
  6. [Dickson et al., 1916]
    L. DICKSON, H.F. BUCHFELDT, AND G.A. MILLER, Theory and applications of finite groups, John Wiley, New York, 1916.zbMATHGoogle Scholar
  7. [Eisenbud and Saltman, 1987]
    D. EISENBUD AND D. SALTMAN, Rank varieties of matrices, Commutative algebra (Berkeley, CA, 1987), 173–212, Math. Sci. Res. Inst. Publ. 15, Springer, New York, 1989.Google Scholar
  8. [Eriksson and Linusson, 1995]
    K. ERIKSSON AND S. LINUSSON, The size of Fulton’s essential set, Sem. Lothar. Combin., 34 (1995), pp. Art. B341, approx. 19 pages (electronic).MathSciNetGoogle Scholar
  9. [Eriksson and Linusson, 2000a]
    K. ERIKSSON AND S. LINUSSON, A combinatorial theory of higher-dimensional permutation array, Adv. in Appl. Math. 25 (2000), no. 2, 194–211.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [Eriksson and Linusson, 2000b]
    K. ERIKSSON AND S. LINUSSON, A decomposition of Fl(n)d indexed by permutation arrays, Adv. in Appl. Math. 25 (2000), no. 2, 212–227.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [Fulton, 1991]
    W. FULTON, Flags, Schubert polynomials, degeneracy loci, and determinantal formulas, Duke Math. J., 65 (1991), pp. 381–420.CrossRefMathSciNetGoogle Scholar
  12. [Fulton, 1997]
    W. FULTON, Young tableaux, with Applications to Representation Theory and Geometry, London Math. Soc. Student Texts 35, Cambridge U.P., Cambridge, 1997.Google Scholar
  13. [Gonciulea and V. Lakshmibai, 2001]
    N. GONCIULEA AND V. LAKSHMIBAI, Flag varieties, Hermann-Actualities Mathématiques, 2001.Google Scholar
  14. [Harris, 1979]
    J. HARRIS, Galois groups of enumerative problems, Duke Math. J. 46 (1979), no. 4, 685–724.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [Hartshome, 1977]
    R. HARTSHORNE, Algebraic Geometry, GTM 52, Springer-Verlag, New York-Heidelberg, 1977.zbMATHGoogle Scholar
  16. [Jordan, 1870]
    C. JORDAN, Traite des Substitutions, Gauthier-Villars, Paris, 1870.Google Scholar
  17. [Kleiman, 1987]
    S. KLEIMAN, Intersection theory and enumerative geometry: a decade in review, in Algebraic geometry, Bowdoin, 1985, Proc. Sympos. Pure Math., 46, Part 2, 321–370, Amer. Math. Soc, Providence, RI, 1987.Google Scholar
  18. [Knutson, 2001]
    A. KNUTSON, Descent-cycling in Schubert calculus, Experiment. Math., 10 (2001), no. 3, 345–353.zbMATHMathSciNetGoogle Scholar
  19. [Knutson and Tao, 2001]
    A. KNUTSON AND T. TAO, Honeycombs and sums of Hermitian matrices, Notices Amer. Math. Soc, 48 (2001), 175–186.zbMATHMathSciNetGoogle Scholar
  20. [Kumar, 2002]
    S. KUMAR, Kac-Moody Groups, Their Flag Varieties and Representation Theory, Progress in Math., 204, Birkhäuser, Boston, 2002.Google Scholar
  21. [Lang, 1983]
    S. LANG, Fundamentals of Diophantine Geometry, Springer-Verlag, New York, 1983.zbMATHGoogle Scholar
  22. [Lascoux and Schützenberger, 1982]
    LASCOUX, A. AND M.-P. SCHÜTZENBERGER, Polynômes de Schubert, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 13, 447–450.zbMATHGoogle Scholar
  23. [Macdonald, 1991]
    I.G. MACDONALD, Notes on Schubert Polynomials, Publ. du LACIM Vol. 6, Université du Québec à Montréal, Montreal, 1991Google Scholar
  24. [Magyar, 2005]
    P. MAGYAR, Bruhat order for two flags and a Line, Journal of Algebraic Combinatorics, 21 (2005).Google Scholar
  25. [Magyar and van der Kallen, 1999]
    P. MAGYAR AND W. VAN DER KALLEN, The Space of triangles, vanishing theorems, and combinatorics, Journal of Algebra, 222 (1999), 17–50.zbMATHCrossRefMathSciNetGoogle Scholar
  26. [Manin, 1974]
    YU. MANIN, Cubic forms: Algebra, Geometry, Arithmetic, North-Holland, Amsterdam, 1974.zbMATHGoogle Scholar
  27. [Manivel, 1998]
    L. MANIVEL, Symmetric Functions, Schubert Polynomials and Degeneracy Loci, J. Swallow trans. SMF/AMS Texts and Monographs, Vol. 6, AMS, Providence RI, 2001.zbMATHGoogle Scholar
  28. [Mnëv, 1985]
    N. MNÈV, Varieties of combinatorial types of projective configurations and convex polyhedra, Dolk. Akad. Nauk SSSR, 283 (6) (1985), 1312–1314.Google Scholar
  29. [Mnëv, 1988]
    N. MNÈV, The universality theorems on the classification problem of configuration varieties and convex polytopes varieties, in Topology and geometry — Rohlin seminar, Lect. Notes in Math. 1346, Springer-Verlag, Berlin, 1988, 527–543.CrossRefGoogle Scholar
  30. [Purbhoo, 2006]
    K. PURBHOO, Vanishing and nonvanishing criteria in Schubert calculus, International Math. Res. Not., Art. ID 24590 (2006), pp.1–38.CrossRefGoogle Scholar
  31. [Serre, 1989]
    J.-P. SERRE, Lectures on the Mordell-Weil theorem, M. Waldschmidt trans. F. Viehweg, Braunschweig, 1989.zbMATHGoogle Scholar
  32. [Shapiro et al., 1997]
    B. SHAPIRO, M. SHAPIRO, AND A. VAINSHTEIN, On combinatorics and topology of pairwise intersections of Schubert cells in SL n/B, in The Amol’d-Gelfand Mathematical Seminars, 397–437, Birkhäuser, Boston, 1997.Google Scholar
  33. [Vakil, 2006a]
    R. VAKIL, A geometric Littlewood-Richardson rule, with an appendix joint with A. Kmitson, Ann. of Math. (2) 164 (2006), no. 2, 371–421.zbMATHCrossRefMathSciNetGoogle Scholar
  34. [Vakil, 2006b]
    R. VAKIL, Schubert induction, Ann. of Math. (2) 164 (2006), no. 2, 489–512.zbMATHCrossRefMathSciNetGoogle Scholar
  35. [Vakil, 2006c]
    R. VAKIL, Murphy’s Law in algebraic geometry: Badly-behaved deformation spaces, Invent. Math. 164 (2006), no. 3, 569–590.zbMATHCrossRefMathSciNetGoogle Scholar
  36. [Weber, 1941]
    H. WEBER, Lehrbuch der Algebra, Chelsea Publ. Co., New York, 1941.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Sara Billey
    • 1
  • Ravi Vakil
    • 2
  1. 1.Department of MathematicsUniversity of WashingtonSeattle
  2. 2.Department of MathematicsStanford UniversityStanford

Personalised recommendations