Evgeny Smirnov's classes

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.

Suggested reading

  1. William Fulton, Young tableaux, with applications to representation theory and geometry. CUP, 1997 (Russian translation available)
  2. Ian G. Macdonald. Symmetric functions and Hall polynomials. 2nd edition. Oxford University Press, 1995 (Russian translation of the 1st edition available).
  3. Laurent Manivel. Fonctions symétriques, polynômes de Schubert et lieux de dégénérescence. SMF, 1999 (English translation available)
  4. Ian G. Macdonald. Notes on Schubert polynomials. UdQ, Montréal, 1991.
  5. Allen Knutson. Schubert polynomials and symmetric functions.
  6. 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).