MA 319A: Schubert Calculus
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Instructor:
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Arvind Ayyer
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Office:
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X-15 (new wing)
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Phone number:
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(2293) 3215
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Email:
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(First name) at iisc dot ac dot in
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Class Timings:
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Tuesdays and Thursdays, 3:30–5:00pm.
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Classroom:
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LH 5, Mathematics Department (second floor)
To join the course on MS Teams, use the code i7hger8
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Office hours:
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By appointment
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Textbook:
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Symmetric functions, Schubert polynomials and degeneracy loci
by L. Manivel,
American Mathematical Society, Providence, RI, 2001
ISBN-13 - 978-0-821821541
Introduction to the Cohomology of the Flag Variety
by Sara C. Billey, Yibo Gao and Brendan Pawlowski
To appear as a chapter in Handbook of Combinatorial Algebraic Geometry: Subvarieties of the Flag Variety
arXiv version
Young tableaux
by W. Fulton
Cambridge University Press, 1997
ISBN-10 - 0-521-56144-2
Schubert polynomials
by Ian Macdonald
Surveys in combinatorics, London Math. Soc. Lecture Note Ser., 166, Cambridge University Press, 1991
ISBN-10 - 0-521-40766-4
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Course Prerequisites
MA 261 or equivalent.
Course Description
This course will provide an elementary introduction to the
combinatorial aspects of Schubert calculus, the part of enumerative
geometry dealing with classical varieties such as Grassmanians, flag
varieties, and their Schubert varieties.
Topics include Grassmann-Plucker relations, Schubert cells, Pieri's
formula, Young tableaux, Schur polynomials, Jacobi-Trudi and
Giambelli's formulas, Littlewood-Richardson rule, Gelfand-Serganova
cells and matroids, Bruhat order, Chevalley-Monk's formula, Schubert
polynomials.
Exams
All exams will be closed book, closed notes, and
no calculators or electronic devices are allowed.
No communication among the students will be tolerated.
There will be no make up exams.
The date for the midterm and final will be announced later.
Grading
Here are the weights for the homework and exams.
All marks will be posted online
on Moodle.
- 15% – Class participation
- 35% – Midterm
- 50% – Final
Tentative Class Plan
Week 1: Projective geometry
Enumerative geometry, projective space, affine patches,
vanishing locus, affine variety, projective variety,
Zariski topology, Grassmannians, row reduced Echelon form
Week 2: Grassmannians
Plucker coordinates, Plucker relations
Week 3: Flag varieties
complete and partial flag varieties,
book picture for flags, canonical matrix,
partitions, Young diagrams, complement,
Young's poset
Week 4: Schubert varieties
Schubert cells and varieties, Bruhat decomposition,
standard and standard opposite flags,
transverse flags, duality theorem,
Week 5: Cellular homology
CW complex, cellular boundary map, chain complex,
cellular homology and cohomology groups,
Poincare duality, Schubert classes
Week 6: Symmetric functions
Ring of symmetric functions, elementary and complete
symmetric functions, standard involution, dominance order
Hall inner product, Schur functions, Kostka numbers.
Week 7: Schur functions
RS and RSK algorithms, Cauchy and dual Cauchy identities
and consquences, bialternant formula
Week 8: Schur functions
Pieri and dual Pieri rule, Littlewood-Richardson rule,
Jacobi-Trudi identities, Frobenius determinant formula,
hook length formula
Week 9: Schubert calculus in the Grassmannian
Pieri rule for Schubert cycles, Littlewood-Richardson rule
for products of Schubert cycles, intersection numbers,
Rothe diagrams, Lehmer codes
Week 10: Schubert calculus in the flag variety
Bruhat decomposition in the flag variety, Bruhat order in the
symmetric group, cohomology ring of the flag variety, coinvariant
algebra, Borel presentation.
Week 11: Schubert polynomials
Monk's formula, special Schubert classes, transition equation for
Schubert polynomials, examples, dominant permutations
Week 12: Relating Schubert classes and polynomials
Schubert polynomials form a basis of the polynomial ring,
Products of Schubert polynomials give the Schubert structure
constants, Monk's formula for Schubert polynomials, divided
differences, Lascoux-Schützenberger's theorem.
Week 13: Combinatorial formulas
Wiring diagrams, pipe dreams and reduced pipe dreams, compatible
pairs, Billey-Jockusch-Stanley theorem, bottom and top pipe dreams,
chute and ladder moves, Billey-Bergeron theorem.
Week 14: Combinatorial formulas
Little's bump algorithm, nearly reduced words, bounded pairs,
bounded bump algorithm, stack push, double Schubert polynomials,
Macdonald's identity.