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.