Schubert Varieties (2017)

The intention was to cover as much of Laurent Manivel’s monograph on the subject. There are three chapters in that monograph and correspondigly there were three courses in the workshop.

Symmetric functions and the representation theory of the symmetric group (the first chapter) was covered by AMRITANSHU PRASAD and SANKARAN VISWANATH. The topic of Schubert polynomials (the second chapter) was covered by VIJAY RAVIKUMAR and K. N. RAGHAVAN. The geometry of Schubert varieties (the third chapter) was covered by SUDHIR GHORPADE and EVGENY SMIRNOV. Three lectures on cohomology and characteristic classes in general were given by PARAMESWARAN SANKARAN.

Listed below in separate subsections are the synopses of the courses in the words of the respective speakers.

Amritanshu Prasad (8 lectures of 90 minutes each).

• Lecture 1
• Symmetric polynomials
• Elementary and complete symmetric polynomials
• Transition from monomial to elementary and complete polynomials
• Lecture 2:
• Alternating polynomials
• Bialternant form of Schur polynomials
• Basis of Schur polynomials
• Lecture 3:
• Abaci and alternants
• Pieri’s formula
• Transition from Schur to elementary and symmetric polynomials
• Semistandard tableaux, Kostka numbers
• Lecture 4:
• Triangularity of Kostka numbers.
• Schensted insertion
• Insertion tableau of a word
• Lecture 5:
• The plactic monoid
• Greene’s theorems
• Section theorem
• Lecture 6:
• Plactic Pieri rules
• Kostka’s definition of Schur functions
• Lecture 7:
• Lindstrom-Gessel-Viennot Lemma
• Jacobi-Trudi identities
• Giambelli’s formula
• Lecture 8:
• RSK correspondence
• The Littlewood-Richardson rule

Sankaran Viswanath (4 lectures of 90 minutes each). Review of the theory of finite dimensional complex representations of finite groups:

• Symmetric polynomials as characters of polynomial representations of GL(V ).
• The Frobenius characteristic map and the relation between the characters of the symmetricgroup and symmetric functions.
• Construction of the irreducible representations of the symmetric group; Specht modules.
• The restriction problem for irreducible representations of symmetric groups.

Vijay Ravikumar (4 lectures of 90 minutes each).

• Lecture 1: Rothe Diagrams, Lehmer codes, essential boxes, rank functions, length of permutation
• generators and relations for Sn, reduced decompositions of permutations
• a canonical reduced decomposition (’Lascoux’s magic trick”)
• the graph of reduced words for a given permutation
• the weak and strong Bruhat orders on Sn
• statement of subword property
• Lecture 2: More on the Bruhat order
• writing the inversion set based on a given decomposition of a permutation
• the strong exchange property, statement and proof
• Bruhat order in terms of key tableaux
• Bruhat order in terms of associated rank functions
• definition of Grassmannian permutations
• Lecture 3: Wiring diagrams, Divided difference operators
• conclusion of proof of subword property
• wiring diagrams for permutations
• proof that the graph of reduced words is connected
• divided difference operators
• definition and uniqueness of Schubert polynomials
• Lecture 4: Computing Schubert polynomials
• existence of Schubert polynomials (i.e. stability under inclusion of Sn into Sn+1)
• computing Schubert polynomials using divided difference operators
• Fomin-Kirillov configurations (i.e. wiring diagrams written a special way)
• computing a Schubert polynomial by resolving crossings in a particular wiring diagram
• Stanley’s conjecture for the Schubert polynomial in terms of compatible sequences

K. N. Raghavan (5 lectures of 90 minutes each). The topics covered were:

• Basic facts about Schubert polynomials: for instance, they form an integral basis for the ring Z[x1, x2, . . .] over the integers in countably infinitely many variables.
• Monk’s rule for the multiplication of a Schubert polynomial with xi
• The nil-Hecke algebra and the Yang-Baxter equation: applications to the computation of double Schubert polynomials: justification for the Fomin-Kirillov procedure and in turn the proof of Stanley’s conjecture.
• Facts about the co-invariant ring for the action of the symmetric group on n letters on the polynomial ring in n variables: for instance, that it is the regular representation (as a module for the group).
• Cauchy’s formula for double Schubert polynomials.

Sudhir Ghorpade (6 lectures of 90 minutes each).

• Lecture 1: Introduction to Grassmannians, Plucker embedding, Grassmannian as a homogeneous space (transitive group action of the General linear group), Plucker relations
• Lecture 2: Sufficiency of Plucker relations, Ideal of the Grassmannian. Schubert cells and Schubert varieties in Grassmannians, Cellular decomposition of the Grassmannian,
• Lecture 3: Cellular decomposition of Schubert varieties. Inclusions of Schubert varietiesin Grassmannians and the Bruhat order, Poincare polynomial of the Grassmannian, Connection with the number of Fq-rational points of the Grassmannian. Schubert varieties as linear sections of Grassmannians.
• Lecture 4: Standard monomials, Hodge basis theorem for the homogeneous coordinate ring of Grassmannians and more generally, for the homogeneous coordinate ring of Schubert varieties in Grassmannians. Applications to the determination of the vanishing ideal of Grassmannians. Hodge postulation formula for Schubert varieties in Grassmannians.
• Lecture 5: Cohomology ring of the Grassmannian (with integer coefficients), Schubert cycles, Cup products, Intersections of Schubert varieties. Cup products of Schubert classes of complementary codimension.
• Lecture 6: Pieri’s formula for Schubert classes. Applications: Relation of Schur functions and Schubert classes (in Grassmannians), Giambelli’s formula, Product rule for Schubert classes.

 Evgeny Smirnov (6 lectures of 90 minutes each). Singularities of Schubert varieties:

• Lecture 1: Schubert cells and Schubert varieties in Grassmannians. Schubert varieties and enumerative geometry. Example: the four lines problem. Bott-Samelson resolutions of singularities of Schubert varieties.• Lecture 2: Bott-Samelson resolutions of singularities of Schubert varieties. Smoothness criterion for Schubert varieties. Poincare polynomial for a Grassmannian, q-binomial coefficients. Small resolutions of singularities.
• Lecture 3: Singular loci of Schubert varieties: Tangent space to a Schubert variety
• Lecture 4: Singular loci of Schubert varieties (continued): Singularity criterion, codimension of the singular locus. Degree of a Schubert variety.
• Lecture 5: Flag varieties. Schubert varieties, Schubert classes, Bruhat order. The Schubert classes form a self-dual basis.
• Lecture 6: Monk’s rule. Recall on Chern classes. Borel’s presentation of the cohomology ring of a full flag variety.

Parameswaran Sankaran (3 lectures of 90 minutes each).

• Lecture 1: Singular homology and cohomology. Cellular (co)homology of a (finite) CW complex. Relation to de Rham cohomology of smooth manifolds. Orientability of a manifold. Ring structure of cohomology and Poincare duality. Cohomology class dual to a  ́submanifold. Cohomology ring of a complex projective space.
• Lecture 2: Cohomological implications of Kahler structure on a compact complex manifold. Holomorphic line bundles and their Chern classes.
• Lecture 3: The cohomology ring of the infinite complex Grassmannian as the ring of symmetric polynomials in several variables. Cohomology ring of complex Grassmann manifolds.

Resource persons and subjects

1. Amritanshu Prasad (AP) Symmetric functions
2. Sankaran Viswanath (SV): symmetric functions (continued)
3. Vijay Ravikumar (VR): Schubert polynomials
4. K. N. Raghavan (KNR): Schubert polynomials (continued)
5. Sudhir Ghorpade (SG): geometry of Schubert varieties
6. Evgeny Smirnov (ES): geometry of Schubert varieties (continued)
7. Parameswaran Sankaran (PS): pre-requisite topological background

Time Table

Link to site:
https://www.imsc.res.in/~knr/past/2017_sch_atmw/index.html

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