Teaching

Math 290Z

Meets: Wednesdays and Fridays during the Spring 2025 semester from 9am to 10:15am at Science Ctr 412.
Office hours: Friday 10:30-11:30 and by appointment.

Course

This course will be an introduction to the combinatorics of symmetric functions at the graduate level. We will introduce the classical families of symmetric functions and some of their generalizations (Macdonald polynomials Jack polynomials,..). We will discuss how these functions encode various combinatorial objects such as tableaux, plane partitions and combinatorial maps.

References

The main references will be:

• R.P. Stanley: Enumerative Combinatorics (Vol. 2)
• I.G. Macdonald: Symmetric functions and Hall polynomials
• J. Haglund, M. Haiman, N. Loehr: A Combinatorial Formula for Macdonald Polynomials
• G. Chapuy, M. Dołęga: Non-orientable branched coverings, b-Hurwitz numbers, and positivity for multiparametric Jack expansions

Prerequisites: Basic linear algebra (such as from Math 122). Familiarity with basic combinatorial notions would be helpful.

Tentative schedule

• Lecture 1 (W Jan 29): Outline of the course
• Lecture 2 (F Jan 31): The space of symmetric functions
• Lecture 3 (W Feb 5): The involution ω, the Hall scalar product
• Lecture 4 (F Feb 7): Schur functions
• Lecture 5 (W Feb 12): RSK algorithm
• Lecture 6 (F Feb 14): The Jacobi Trudi identity
• Lecture 7 (W Feb 19): The Pieri rule, the determinant formula of Schur functions
• Lecture 8 (F Feb 21): The Murnaghan-Nakayama rule
• Lecture 9 (W Feb 26): Characters of the symmetric group
• Lecture 10 (F Feb 28): Plane partitions
• Lecture 11 (W Mar 5): Quasisymmetric functions
• Lecture 12 (F Mar 7): Plethystic notation
• Lecture 13 (W Mar 12): Superization
• Lecture 14 (F Mar 14): Macdonald polynomials
• Lecture 15 (W Mar 26): Haglund-Haiman-Loehr formula
• Lecture 16 (F Mar 28): Haglund-Haiman-Loehr formula continued
• Lecture 17 (W Apr 2): Guest lecture by Lauren Williams: Macdonald polynomials and multiline queues
• Lecture 18 (F Apr 4): Orientable maps
• Lecture 19 (W Apr 9): The Generating function of orientable Maps
• Lecture 20 (F Apr 11): Young symmetrizers and the Stanley-Féray formula
• Lecture 21 (W Apr 16): Non-orientable maps
• Lecture 22 (F Apr 18): Goulden-Jackson conjectures
• Lecture 23 (W Apr 23): Chapuy-Dołęga formula for the mutliparametric series of Jack polynomials
• Lecture 24 (F Apr 25): Chapuy-Dołęga formula for the mutliparametric series of Jack polynomials
• Lecture 25 (W Apr 30): Final project presentation