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.

Office hours: by appointment.

# 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.