Math 206A: Symmetric Functions (Fall 2022)

Course description: Roughly half of the material will be based on Stanley's Enumerative Combinatorics, vol. 2, chapter 7. We will then discuss some of the more advanced topics. Here is a partial list:- Young diagrams, Young tableaux, Schur functions, hook-length formula.
- RSK algorithm, Cauchy identity.
- Jeu de taquin, Knuth equivalence, Fomin's growth diagrams.
- Littlewood-Richardson rule, crystal graphs, Knutson-Tao puzzles.
- Connections to representation theory and algebraic geometry.
- Schubert polynomials, pipe dreams.
- q,t-Catalan numbers, diagonal harmonics, shuffle conjecture, Macdonald polynomials.

Instructor: Pavel Galashin (udе.аlсu.htаm@nihsаlаg). Please put "206A" into the subject line.

Time and location: MWF 4pm-4:50pm, MS 5118.

Grading: Based on several homework problem sets.

Prerequisites: The material will be accessible to first year graduate students and advanced undergraduates. Knowledge of linear algebra and some prior exposure to combinatorics (e.g. Catalan numbers) would be very helpful.

# Homeworks

- Homework #1, due on Gradescope on Monday, October 10, at 4pm.
- Homework #2, due on Gradescope on Monday, October 17, at 4pm.
- Homework #3, due on Gradescope on Monday, October 24, at 4pm.
- Homework #4, due on Gradescope on Monday, October 31, at 11:59pm.
- Homework #5, due on Gradescope on Monday, November 7, at 4pm.
- Homework #6, due on Gradescope on Monday, November 14, at 4pm.
- Homework #7, due on Gradescope on Wednesday, November 23, at 4pm.

# Sage worksheets

- Basics of Sym (10/10/22) gives examples of computing with the symmetric functions that we have studied so far. See also Sage reference manuals on partitions and symmetric functions; there's also this tutorial.

# Lectures

- Lecture 1 (09/23/22). Logistics, three random facts (partitions, standard Young tableaux, lozenge tilings). Definition and basic examples of symmetric polynomials.
- Lecture 2 (09/26/22). Schur polynomials. The ring of symmetric polynomials. Symmetric functions: definition, examples, dimensions of graded pieces.
- Lecture 3 (09/28/22). Fundamental theorem of symmetric functions (Λ is a polynomial ring in the elementary symmetric polynomials). Complete homogeneous symmetric functions.
- Lecture 4 (09/30/22). The omega involution.
- Lecture 5 (10/03/22). Background on generating functions. The exponential formula.
- Lecture 6 (10/05/22). The scalar product on Λ.
- Lecture 7 (10/07/22). Power sum symmetric functions.
- Lecture 8 (10/10/22). Power sum continued. Summary of properties of m, e, h, p, omega involution, and the scalar product.
- Lecture 9 (10/12/22). Schur functions. (Form a basis.)
- Lecture 10 (10/14/22). RSK algorithm.
- Lecture 11 (10/17/22). Cauchy identity.
- Lecture 12 (10/19/22). Symmetry of the RSK algorithm.
- Lecture 13 (10/21/22). Dual RSK algorithm. Antisymmetric polynomials.
- Lecture 14 (10/24/22). Classical definition of Schur functions.
- Lecture 15 (10/26/22). Classical definition of Schur functions - proofs, skew Schur functions.
- Lecture 16 (10/28/22). Jacobi–Trudi identity statement. Totally positive matrices, Lindström–Gessel–Viennot (LGV) lemma statement.
- Lecture 17 (10/31/22). Proof of LGV and Jacobi–Trudi. Summary.
- Lecture 18 (11/02/22). Summary. Knuth equivalence, statements.
- Lecture 19 (11/04/22). Knuth equivalence, proofs.
- Lecture 20 (11/07/22). Jeu de taquin and Knuth equivalence.
- Lecture 21 (11/09/22). Jeu de taquin. Evacuation and Schützenberger Involution.
- Lecture 22 (11/14/22). Growth diagrams and jeu de taquin.
- Lecture 23 (11/16/22). The Littlewood–Richardson rule.