Symmetric functions, Fall 2019
News and announcements
There will be a regular class (lecture+exercise class) on October 23.
The midterm exam: October 30, from 15:30 to 18:30, room 109.
Lectures and exercise classes
Wednesdays, 15:30-16:50 (lecture, room 427) and 17:00-18:20 (exercise class, room 109).
Lecture 1 (11/09/19). Symmetric polynomials: definition. Bases in the ring of symmetric polynomials: monomial, complete, elementary symmetric polynomials, Newton power sums. Problem set 1.
Lecture 2 (18/09/19). Skew-symmetric polynomials, Vandermonde determinant. Symmetric functions: inverse limit of rings of symmetric polynomials. Problem set 2.
Lecture 3 (25/09/19). The Jacobi-Trudi identity. Pieri’s formulas. e-h involution. Problem set 3.
Lecture 4 (02/10/19). Young tableaux. Relation to symmetric functions. Littlewood’s theorem. Lindström-Gessel-Viennot determinantal formula. Problem set 4.
Lecture 5 (09/10/19). Kostka numbers. The Cauchy product and the Cauchy determinant. Problem set 5.
Lecture 6 (16/10/19). The scalar product, orthogonality of Schur functions. The first and the second Cauchy identities. Problem set 6.
Lecture 7 (23/10/19). The Frame-Robinson-Thrall formula (aka the hook length formula). Principal specialization of a Schur polynomial. MacMahon’s formula(s). No problem set.
Lecture 8 (6/11/19). Arrays (after Danilov and Koshevoy). Operations on arrays. Condensation. No problem set.
Lecture 9 (13/11/19). Arrays, continued. D-, L- and DL-condensed arrays. Row-scan, bijections between dense arrays, Young tableaux, and Yamanouchi words. Fiber product theorem. Problem set 9.
Lecture 10 (20/11/19). Fiber product theorem (cont’d). Symmetric group action, DU-orbits. Schur polynomials as sums over DU-orbits. Problem set 10.
Lecture 11 (27/11/19). Littlewood-Richardson rule. Symmetric group, the Bruhat order, reduced decompositions. Problem set 11.
Lecture 12 (4/12/19). Schubert polynomials. Divided difference operators, linear independence of Schubert polynomials. Problem set 12.
Lecture 13 (11/12/19) The Bruhat order. Monk’s rule. Lascoux transition formula. Positivity of coefficients. Exam preparation problem set.
MIDTERM EXAM: October 30, 2019, room 109.
FINAL EXAM: December 18, 2019.
This is an open-book exam: you are allowed to bring and use any paper sources (books, printed or handwritten notes etc). No laptops/tablets/kindles/phones are allowed.
- William Fulton, Young tableaux, with applications to representation theory and geometry. CUP, 1997 (Russian translation available)
- Ian G. Macdonald. Symmetric functions and Hall polynomials. 2nd edition. Oxford University Press, 1995 (Russian translation of the 1st edition available).
- Laurent Manivel. Fonctions symétriques, polynômes de Schubert et lieux de dégénérescence. SMF, 1999 (English translation available)
- Ian G. Macdonald. Notes on Schubert polynomials. UdQ, Montréal, 1991.
- Allen Knutson. Schubert polynomials and symmetric functions.
- Alexey Gorodentsev. Algebra II. Textbook for students of Mathematics. Springer, 2017. (Russian translation available, material of Lectures 8-10 is covered in Sec.27 of the Russian edition).