18.217 Combinatorial Theory: Symmetric Group and Symmetric Functions

Fall 2020, MIT

Instructor: Alex Postnikov
Grader: Pakawut (Pro) Jiradilok    
Class meetings: MWF 1-2 pm    
Office hours: email for appointments

Zoom: https://mit.zoom.us/j/95569469250

The zoom password is the number of Young diagrams with 4 boxes, written as a number. You can also contact the instructor by email to find the password.

All lectures will be given in real time on zoom. The lectures will be recorded and posted on this webpage. So you can watch them later if your time zone makes it hard to participate in real time. (Video recordings will be access-controlled and limited to the class.) But I encourage everybody to participate in real-time lectures, if possible. At any moment during zoom sessions, please feel free to unmute yourself and ask a question or make a comment. Your feedback (questions, reactions, etc.) is a crucial ingredient of online learning experience. I may not be able to monitor the Chat window during lectures; but please feel free to use Chat; other students in class might be able to answer your questions.

Course description:

The course will be about combinatorics of the symmetric group and symmetric functions. We will discuss Young tableaux, Schur functions, the Robinson-Schensted-Knuth correspondence, the Cauchy identity, the Jacobi-Trudi identity, the hook-length formula, the Littlewood-Richardson rule, the Murnaghan-Nakayama rule, Schutzenberger's involution, jeu de taquin, Hillman-Grassl correspondence, Fomin's growth diagrams, Gelfand-Tsetlin patterns, Berenstein-Zelevinsky's triangles, Knutson-Tao's honeycombs, Jucys-Murphy elements, Hecke algebras, Okounkov-Vershik's construction, Macdonald polynomials, etc.

We will cover classical results in this subject and (as time allows) some recent research advances.

Course Level: Graduate.     The course should be accessible to first year graduate students.

Grading: Based on several problem sets.

Lectures:

You can view all lecture notes and stream video recordings of zoom sessions on canvas.mit.edu.

1. W 09/02/2020. Introduction. Symmetric functions. Young diagrams. Fundamental theorem of symmetric functions. Notes    

2. F 09/02/2020. Relations between elementary, complete homogeneous, and power symmetric functions. The involution omega. Notes    

   M 09/07/2020. Labor Day - holiday.

3. W 09/09/2020. Schur symmetric functions. Determinant formula. Semi-standard Young tableaux. Gelfand-Tsetlin patterns. Notes    

4. F 09/11/2020. Symmetric group. Wiring diagrams and reduced decompositions. Schur polynomials as Schubert polynomials and Demazure characters. Notes    

5. M 09/14/2020. Permutohedra. Divided differences vs Demazure operators. Pieri rules. Notes

6. W 09/16/2020. Cauchy identities. Robinson-Schensted-Knuth correspondence (RSK). Notes

7. F 09/18/2020.

8. M 09/21/2020.

9. W 09/23/2020.

10. F 09/25/2020.

11. M 09/28/2020.

12. W 09/30/2020.

13. F 10/02/2020.

14. M 10/05/2020.

15. W 10/07/2020.

16. F 10/09/2020.

    M 10/12/2020. Columbus Day - holiday.

17. Tuesday 10/13/2020. (Monday schedule of classes)

18. W 10/14/2020.

19. F 10/16/2020.

20. M 10/19/2020.

21. W 10/21/2020.

22. F 10/23/2020.

23. M 10/26/2020.

24. W 10/28/2020.

25. F 10/30/2020.

26. M 11/02/2020.

27. W 11/04/2020.

28. F 11/06/2020.

29. M 11/09/2020.

    W 11/11/2020. Veterans Day - holiday.

30. F 11/13/2020.

31. M 11/16/2020.

32. W 11/18/2020.

33. F 11/20/2020.

    M 11/23/2020. Thanksgiving vacation.

    W 11/25/2020. Thanksgiving vacation.

    F 11/27/2020. Thanksgiving vacation.

34. M 11/30/2020.

35. W 12/02/2020.

36. F 12/04/2020.

37. M 12/07/2020.

38. W 12/09/2020. Last day of classes.

Recommended textbooks:

• I. G. Macdonald, Symmetric Functions and Hall Polynomials: pdf.
• Richard P. Stanley, Enumerative Combinatorics, Volume 2, Chapter 7: Symmetric Functions.
• Bruce E. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions: pdf.

• 18.315 Combinatorial Theory, MIT, Fall 2006.

• 18.315 Combinatorial Theory, MIT, Fall 2010.

This webpage will be updated periodically. All information related to the course (lecture notes, recordings of zoom lectures, problem sets, etc.) will be posted here.

last updated: September 14, 2020