Schubert variety
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| Schubert variety | |
|---|---|
| Name | Schubert variety |
| Field | Algebraic geometry |
| Introduced | 19th century |
| Notable | Hermann Schubert, Ehresmann, Bruhat |
Schubert variety is a type of subvariety arising in algebraic geometry and representation theory associated to flag varieties and Grassmannians. Defined combinatorially via Weyl groups and Bruhat order, these varieties connect to classical topics such as intersection theory, cohomology rings, and singularity theory while appearing in contexts including enumerative geometry, geometric representation theory, and Schubert calculus. Key figures linked to their development include Hermann Schubert, Élie Cartan, Claude Chevalley, and Alexander Grothendieck.
Definition and basic properties
A Schubert variety is a projective subvariety of a flag variety G/B or a partial flag variety G/P for a complex reductive group G, defined as the closure of a Bruhat cell indexed by an element w of the Weyl group W. Important structural properties relate to the Bruhat order on W, the action of a maximal torus T in G, and the Borel subgroup B, with links to the representation theory of Lie algebras and the geometry of homogeneous spaces such as Grassmannians and full flag varieties. Basic invariants include dimension given by the length ℓ(w) in W, codimension, and descriptions via incidence conditions in classical cases like GL_n and SL_n flag manifolds.
Schubert cells and Bruhat decomposition
The Bruhat decomposition expresses G as a union of double cosets BwB indexed by W, yielding Schubert cells BwB/B inside G/B that are isomorphic to affine spaces; closures of these cells are Schubert varieties. The combinatorics of reduced expressions in W, Coxeter generators such as simple reflections s_i, and pattern avoidance in permutations in the symmetric group S_n govern cell structure and closure relations, tying to the geometry of flag manifolds and the study of Bott–Samelson varieties. Connections appear with classical objects like Young tableaus, Bruhat order, and the Peterson variety in type A and beyond.
Cohomology and intersection theory
Cohomology classes of Schubert varieties form additive bases of the cohomology ring H*(G/B) and the Chow ring A*(G/P), underpinning Schubert calculus. Structure constants—Littlewood–Richardson coefficients in type A—encode intersection numbers and multiplicative relations; these constants relate to combinatorial rules involving Littlewood–Richardson rule, Young diagrams, and symmetric function theory such as Schur functions and Schubert polynomials. Equivariant cohomology with respect to T yields refinements studied via localization theorems of Berline–Vergne and Atiyah–Bott, while quantum cohomology introduces Gromov–Witten invariants connecting to mirror symmetry and enumerative problems on rational curves in flag varieties.
Singularities and desingularizations
Many Schubert varieties are singular; their singular loci are described using pattern avoidance in permutations and criteria from Kazhdan–Lusztig theory. Kazhdan–Lusztig polynomials and intersection cohomology provide invariants detecting rational singularities and perversity conditions relating to Beilinson–Bernstein localization and category O in representation theory. Desingularizations arise via Bott–Samelson resolutions and Demazure operators, with links to the geometry of Springer fibers, the Springer correspondence, and work of Michel Demazure and Hans Samelson. Normality, Cohen–Macaulayness, and Frobenius splitting properties have been established using methods from Geometric Invariant Theory and representation-theoretic techniques.
Examples and classification
Classical examples include Schubert varieties in the Grassmannian Gr(k,n) corresponding to Young diagrams and incidence conditions in projective spaces like CP^n; these recover classical enumerative problems studied by Hermann Schubert and later formalized by Giambelli and Pieri. In type A, combinatorial classification uses permutations in S_n and pattern avoidance criteria of Lakshmibai–Sandhya. Other types involve Weyl groups of Lie types B, C, D and exceptional types, with phenomena reflected in orthogonal and symplectic Grassmannians and isotropic flag varieties linked to groups such as SO_n and Sp_{2n}.
Applications and generalizations
Schubert varieties appear in geometric representation theory (e.g., the geometric Satake correspondence, perverse sheaves on G/B), combinatorics (symmetric functions, Young tableau combinatorics), and mathematical physics (quantum cohomology, integrable systems). Generalizations include Richardson varieties (intersections of Schubert and opposite Schubert varieties), positroid varieties in the Grassmannian related to total nonnegativity and cluster algebras, and affine Schubert varieties in affine Grassmannians and affine flag varieties tied to loop groups and Kac–Moody algebras. Research continues linking them to moduli spaces such as Quiver varietys, canonical bases of quantum groups, and categorical structures in Langlands program contexts.