9.4 Flag varieties and Schubert calculus

Flag varieties and Schubert calculus are key concepts in Lie group actions and homogeneous spaces. They provide a geometric framework for understanding the structure of certain algebraic varieties and their relationships to Lie groups.

Schubert calculus, in particular, offers powerful tools for solving problems in enumerative geometry. By studying intersections of Schubert varieties, we can tackle complex counting problems and gain insights into the cohomology of flag varieties.

Flag Varieties: Definition and Structure

Definition and Relationship to Lie Groups and Homogeneous Spaces

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• Flag varieties are algebraic varieties that parametrize certain sequences of subspaces of a vector space called flags
• A flag in an n-dimensional vector space V is a sequence of subspaces {0} = V₀ ⊂ V₁ ⊂ ... ⊂ Vₙ = V, where each Vᵢ has dimension i
• The flag variety F(d₁, ..., dₖ; n) parametrizes flags where the subspaces have dimensions d₁, ..., dₖ
• Flag varieties are homogeneous spaces for certain linear algebraic groups, such as the general linear group GL(n) or the special linear group SL(n)

Complete and Partial Flag Varieties

• The flag variety F(1, 2, ..., n-1; n) is the quotient space GL(n)/B, where B is the subgroup of upper triangular matrices
  • This is an example of a complete flag variety
• Partial flag varieties, such as the Grassmannian, parametrize flags with subspaces of specific dimensions
  • These are quotients of GL(n) by certain parabolic subgroups
• Examples of partial flag varieties include the Grassmannian G(k, n), which parametrizes k-dimensional subspaces of an n-dimensional vector space, and the incidence variety I(p, q; n), which parametrizes pairs of subspaces with dimensions p and q that intersect in a specific way

Schubert Cells and Decomposition

Schubert Cells and Their Properties

• Flag varieties admit a cell decomposition into Schubert cells, which are indexed by elements of the corresponding Weyl group (e.g., the symmetric group Sₙ for GL(n))
• Schubert cells are orbits of the Borel subgroup B acting on the flag variety
  • They are locally closed subvarieties isomorphic to affine spaces
• The dimension of a Schubert cell is given by the length of the corresponding Weyl group element
• Example: In the complete flag variety F(1, 2, ..., n-1; n), the Schubert cell corresponding to the permutation w ∈ Sₙ is the set of flags (V₁, ..., Vₙ₋₁) such that dim(Vᵢ ∩ F_j) = #{k ≤ i | w(k) ≤ j} for all i, j, where F_• is a fixed reference flag

Schubert Varieties and CW Complex Structure

• The closure of a Schubert cell is called a Schubert variety
  • Schubert varieties are irreducible subvarieties of the flag variety
• Schubert cells form a CW complex structure on the flag variety, with the Schubert varieties as the closures of the cells
• The intersection of two Schubert varieties is a union of Schubert varieties, and the multiplicities of the intersection are given by certain structure constants
• Example: In the Grassmannian G(2, 4), the Schubert variety corresponding to the partition (2, 1) is the set of 2-dimensional subspaces that intersect a fixed 2-dimensional subspace in at least a 1-dimensional subspace

Schubert Calculus: Applications

Intersection Theory and Cohomology

• Schubert calculus is the study of the intersection theory of Schubert varieties in flag varieties
• The cohomology ring of a flag variety has a basis given by the classes of Schubert varieties, called Schubert classes
• The intersection product of Schubert classes corresponds to the intersection of Schubert varieties in the flag variety
• Structure constants for the intersection product of Schubert classes are called Littlewood-Richardson coefficients

Enumerative Geometry Problems

• Schubert calculus can be used to solve enumerative geometry problems, such as counting the number of lines satisfying certain incidence conditions with respect to given subspaces
• Example: How many lines in ℙ³ intersect four given lines? This can be solved by intersecting appropriate Schubert varieties in the Grassmannian G(2, 4)
• Pieri's formula and the Littlewood-Richardson rule provide combinatorial methods for computing intersection numbers and structure constants in the cohomology ring
• Example: Pieri's formula states that the product of a special Schubert class with an arbitrary Schubert class is a sum of Schubert classes indexed by elements obtained by adding a single box to the Young diagram corresponding to the original Schubert class

Fundamental Theorems of Flag Varieties and Schubert Calculus

Pieri Rule and Littlewood-Richardson Rule

• The Pieri rule describes the intersection product of a special Schubert class (corresponding to a simple transposition in the Weyl group) with an arbitrary Schubert class
  • It states that the product is a sum of Schubert classes indexed by elements obtained by adding a single box to the Young diagram corresponding to the original Schubert class
• The Littlewood-Richardson rule gives a combinatorial formula for the structure constants of the cohomology ring of a flag variety, known as Littlewood-Richardson coefficients
  • It involves counting the number of skew Young tableaux satisfying certain conditions, such as the Yamanouchi word condition or the lattice permutation condition

Geometric Techniques and Other Important Theorems

• These rules can be proven using geometric techniques, such as intersecting Schubert varieties with special divisors or using the jeu de taquin algorithm on Young tableaux
• Other important theorems in Schubert calculus include the Giambelli formula, which expresses arbitrary Schubert classes in terms of special Schubert classes, and the Monk formula, which generalizes the Pieri rule to multiplication by classes corresponding to arbitrary simple reflections
• Example: The Giambelli formula expresses a Schubert class σ_λ in the cohomology of the Grassmannian G(k, n) as a determinant of special Schubert classes σ_(1^i), where λ is a partition and (1^i) denotes the partition with i parts equal to 1