Math 665: Schubert Calculus

Winter 2002

Course meets: TTh 11:30-1, 2866 East Hall.

Instructor: Sergey Fomin, 2858 East Hall, 764-6297,

Course homepage:

Level: introductory graduate.

Prerequisites: none (for graduate students).

Student work expected: several problem sets.

Hermann Csar Hannibal Schubert (1848-1911)

This course will provide an elementary introduction to the combinatorial aspects of Schubert calculus, the part of enumerative geometry dealing with such classical varieties as Grassmannians and flag manifolds.

A classical example of a Schubert calculus question is the following: Given mp subspaces of dimension p in general position in a complex vector space of dimension m+p, how many subspaces of dimension m intersect all these mp subspaces nontrivially?

To be able to answer such questions, one needs to gain a concrete understanding of the structure of the cohomology ring of the corresponding variety (in the example above, it will be the Grassmannian Gr(m,m+p)). Computing the intersection numbers like the one defined above requires development of an extensive combinatorial machinery involving Young tableaux, Bruhat orders of finite Coxeter groups, and related aspects of the theory of symmetric functions and Schubert polynomials.

More advanced topics will include (time-permitting): intersection-theoretic computations in (partial) flag manifolds related to classical semisimple Lie groups; quantum cohomology rings and calculation of Gromov-Witten invariants; K-theoretic analogues; and real Schubert calculus.

The presentation will be essentially self-contained and elementary, and will require no special background in combinatorics, topology, algebraic geometry, commutative algebra, or Lie theory.

Course Outline
I. Schubert calculus on Grassmannian manifolds.
II. Schubert calculus on flag manifolds. Schubert polynomials.
III. Variations on the theme of Schubert.

Texts. The course will not strictly follow a particular text. Principal sources:
[F] W.Fulton, Young tableaux , Cambridge University Press, 1997.
[H] J.E.Humphreys, Reflection groups and Coxeter groups, Cambridge University Press, 1994.
[M] L.Manivel, Symmetric Functions, Schubert Polynomials and Degeneracy Loci, AMS, 2001.

Part I: Grassmannians

Lecture 1. Course overview. Grassmannians. Plücker embedding. [M.3.1]
Lecture 2. Grassmann-Plücker relations. Schubert cells and varieties in a Grassmannian. [M.3.1-3.2.1]
Lecture 3. Schubert cells and varieties in a Grassmannian (cont.). [M.3.2.1]
Lecture 4. Cohomology of algebraic varieties. [F.A2] [M.A]
Lecture 5. Duality theorem for Grassmannians. [F.9.4]
Lecture 6. Pieri's formula. Geometric applications. [F.9.4] [M.3.2.2]
Lecture 7. Degree of a Grassmannian. Schur polynomials.
Lecture 8. Combinatorial formula for the Schur polynomials. The ring of symmetric functions.
Lecture 9. Jacobi-Trudi identity.
Lecture 10. Giambelli's formula. Cohomology ring of a Grassmannian.
Lecture 11. Littlewood-Richardson rule.
Lecture 12. K-theory of Grassmannians (guest lecture by Anders Buch).
Lecture 13. Jeu de taquin. Berenstein-Zelevinsky triangles.

    Problem Set 1 (submit 5)
  1. Prove Muir's "law of extensible minors".
  2. Prove that all Plücker coordinates are uniquely determined via the Grassmann-Plücker relations from those labelled by the subsets of the form {1,...,r}-{p}+{q}, where p<=r< q, assuming that P{1,...,r} does not vanish.
  3. Prove that any Schubert variety is the closure of the corresponding Schubert cell.
  4. Prove that a Schubert variety labelled by µ is smooth if and only if µv is a rectangle.
  5. Let V be a subspace of E, and let F and F' be opposite flags in E. Assume that the intersections of V with Fk and F'dim(E)-k have dimensions adding up to dim(V), for any k. Then V is a span of a subset of the standard basis.
  6. Prove that a Schubert variety of codimension 1 (with respect to an arbitrary flag) is the intersection of the Grassmannian with a hyperplane in the Plücker embedding.
  7. Prove that the degree of a Schubert variety labelled by µ is equal to the number of standard Young tableaux of shape µv.
  8. [F], Exercise 18, page 152.
  9. (Same as the "second" Jacobi-Trudi formula below.) [F], Exercise 19, page 152.
  10. [F], Exercise 20, page 152.
  11. [F], Exercise 21(a), page 152.
  12. [F], Exercise 21(b), page 152.
  13. Prove the "second" Pieri formula for the product of a Schur function by an elementary symmetric function.
  14. Prove the "second" Jacobi-Trudi formula, expressing a Schur function in terms of the elementary symmetric functions.
  15. Prove that the power sum symmetric functions pµ form a basis of the ring of symmetric functions over the rational numbers.
  16. Prove that the cohomology ring of the Grassmannian of r-dimensional subspaces in an (r+n)-dimensional space is canonically isomorphic to the quotient of the polynomial ring Z[e1,...,er, h1,...,hn] by the ideal generated by the coefficients of the polynomial (1+e1t+e2t2+...+ ertr) (1-h1t+h2t2-...+ (-1)n hntn)-1.
  17. Prove the equivalence of the classical version of the Littlewood-Richardson rule to the following one: the Littlewood-Richardson coefficient cabp is equal to the number of semistandard Young tableaux T of shape b such that a+weight(T)=p, and furthermore a+weight(T >j) is a partition for all j. (Here T >j denotes the tableau formed by the columns >j of T.)
  18. Prove that the Littlewood-Richardson coefficient cabp is equal to the number of semistandard Young tableaux of shape axb and weight p whose reverse reading word is a ballot sequence. Here axb denotes the skew shape obtained by placing a southwest of b.
  19. Compute the K-theoretic structure constant cabp for a=b=(2,1) and p=(3,3,1).
  20. Prove that the stable Grothendieck polynomial, defined as a generating function for semistandard set-valued tableaux, is a symmetric power series.
  21. Prove that the Grothendieck ring of a Grassmannian is isomorphic to its cohomology ring as an abstract ring.
  22. Prove the fundamental theorem of jeu de taquin (the uniqueness of rectification).
  23. Prove the jeu de taquin version of the Littlewood-Richardson rule.
  24. Prove the Littlewood-Richardson rule formulated in terms of Berenstein-Zelevinsky triangles.

Part II: Flag manifolds

Lecture 14. Flag manifolds and their Schubert cells.
Lecture 15. Schubert varieties in flag manifolds. The Bruhat order.
Lecture 16. Duality theorem for flag manifolds. The Monk-Chevalley formula.
Lecture 17. Reduced words. Divided differences.
Lecture 18. Symmetrization operators. Coinvariant algebra.
Lecture 19. Schubert polynomials.
Lecture 20. Combinatorial formula for the Schubert polynomials.
Lecture 21. Schubert polynomials as a basis in the polynomial ring.
Lecture 22. Algebraic proof of the Monk-Chevalley formula. Borel's theorem.
Lecture 23. The Bernstein-Gelfand-Gelfand-Demazure theorem. Double Schubert polynomials.
Lecture 24. The Cauchy formula for Schubert polynomials. Interpolation.
Lecture 25. Macdonald's identity. Stable Schubert polynomials.
Lecture 26. 321-avoiding permutations. Flagged Schur functions.
Lecture 27. Enumeration of reduced words. Stanley's formula.

    Problem Set 2 (submit 3)
  1. Prove that a matrix r=(rpq) defines a rank diagram for some pair of complete flags in Cn if and only if there exists a permutation w in Sn such that each rpq equals the number of pairs (i,j) with i<=p, j<=q, and j=w(i).
  2. Prove that if the intersection of a Schubert cell labelled by u and a dual Schubert cell labelled by v is non-empty, then u>=v in the Bruhat order.
  3. Prove the geometric version of the Monk-Chevalley formula.
  4. Prove the Tits lemma for the symmetric group.
  5. For any finite reflection group, the corresponding divided difference operators satify the braid relations of appropriate type. Verify this statement for the hyperoctahedral group (types B and C).
  6. Identify the leading monomial of a Schubert polynomial in terms of the code of the corresponding permutation. (State accurately and prove.)
  7. (Counts as 2 problems.) Let In be the ideal in the polynomial ring Z[x1,...,xn] generated by the symmetric polynomials without constant term. Prove that the complete homogeneous polynomials hn(x1), hn-1(x1,x2),..., h1(x1,...,xn) belong to In, generate In, and form a Gröbner basis of In with respect to a suitable monomial order. Prove that the span of Schubert polynomials corresponding to the permutations in Sn is the space of normal forms with respect to this Gröbner basis.
  8. [M], Exercise 2.5.13, page 83. (Newton's interpolation formula.)
  9. Prove the q-analog of Macdonald's identity (for wo).
  10. Let u be a permutation in Sn, and let v be a permutation that fixes 1,...,n. Then the (double) Schubert polynomial for uv is the product of the (double) Schubert polynomials for u and v, respectively.
  11. Prove that any stable Schubert polynomial Gw is Schur-positive. [Difficult]
  12. Prove that a permutation w is 321-avoiding if and only if any two reduced words for w are related via 2-moves.
  13. Let w be a 321-avoiding permutation. Prove that Gw is a skew Schur function for a shape obtained by permuting rows and columns in the diagram of w.
  14. Prove that the number of reduced words for a Grassmannian permutation is equal to the number of standard Young tableaux of the corresponding shape.