Math 206A Algebraic Combinatorics
This is the webpage for Math 206A Fall 2025. If you are not officially enrolled and want to join the mailing-list of this class please email me.
Course description
Instructor: Sylvester Zhang. sylvesterzhang@math.ucla.edu (Please put "206A" into the subject line)
Office hours: Monday 1-2 & by appointment (in general I am available for short discussions after each class).
Topics: Symmetric functions, tableaux combinatorics, and their connections to representation theory. Here's a tentative list of topics we will cover:
- Symmetric functions.
- Tableaux Combinatorics, RSK, etc.
- Representation theory of $S_n$ and ${GL}_n$
- Hall-Littlewood, Kostka polynomials
- Kazhdan-Lusztig polynomials and cells for $S_n$
- Flag varieties, nilpotent orbits, Springer fibers
prerequisites: Linear algebra. The material will be accessible to first year graduate students. Knowledge of abstract algebra (finite groups) will be helpful.
Grading: Based on seven homework assignments (70%), a 50 minutes exam (20%) and participation (10%). Students may choose to give a paper presentation in place of two homework assignments (a list of suggested papers will be posted later). Collaboration on homework assignments is encouraged. Gradeline: A: 90+ B: 80+ C: 70+ D:60+ F:50-
Textbook
The main textbooks are
- Stanley - Enumerative Combinatorics Volume 2 (ch 7)
- Sagan - Symmetric Group.
Other recommended resources:
- Macdonald - Symmetric functions and Hall Polynomials
- Bjorner & Brenti - Coxeter Groups
- TBD
Notes
Lecture notes will be posted here.Schedule
| Date | Topics | |
|---|---|---|
| 9/26/W | Introduction | |
| 9/29/M | Partitions, Dominance order, Young's lattice, up/down operators (Weyl algebra) | HW P1.1 P1.2 posted |
| 10/1/W | $e_\lambda$, $h_\lambda$, and basic properties, FTSP | HW P1.4 posted |
| 10/3/F | generating functions | HW 1 posted |
| 10/6/M | $p_\lambda$, $s_\lambda$, Bender-Knuth involution | |
| 10/8/W | $\omega$-involution, Hall inner product | |
| 10/10/F | Cauchy identity, Schur operators, RS | HW1 due, HW2 posted |
| 10/13/M | RS continued, Fomin-Viennot growth diagrams | |
| 10/15/W | Greene-Kleitman theory | |
| 10/17/F | Geometric detour begins. $G/B$, nilpotent orbits. | |
| 10/20/M | Springer fibers, Steinberg-Spaltenstein theorem. | |
| 10/22/W | Algebraic proof of Greene-Kleitman duality | |
| 10/24/F | Schubert calculus, Borel isomorphism, Grassmanian permutations | |
| 10/27/M | Divided difference, Schubert polynomials, Schur polynomials revisited | |
| 10/29/W | Richardson classes and skew Schur polynomials. Litlewood Richardson rule. End of Geometric detour. | |
| 10/31/F | Room for push back | |
| 11/3/M | Room for push back | |
| 11/5/W | Rep theory begins. Rep theory of finite groups. | |
| 11/7/F | examples, indecomposability, Mascheke's theorem. |
Homework
The homework assignments can be found here. The file will be updated periodically.