Math 665: Combinatorial Theory II

Topic for Fall 2015: Symmetric functions.

Course meets: TuTh 11:40-1:00, Room 1866 East Hall.

Instructor: Sergey Fomin, 4868 East Hall, 764-6297,

Grader: Benjamin Branman,

Office hours: Tuesday 1-2, Thursday 4-6 in 4868 East Hall.

Course homepage:

Level: introductory graduate.

Prerequisites: none (for graduate students).

Student work expected: several problem sets.

Synopsis: This is an introduction to the foundations of the classical theory of symmetric functions from a combinatorial perspective. Core topics include Young tableaux, Schur functions, and related combinatorial algorithms and enumeration problems. The course will conclude by a survey of applications of symmetric functions to various areas of mathematics such as linear algebra, representation theory, and enumerative geometry.

[EC2]   R. P. Stanley, Enumerative combinatorics, vol. 2, Cambridge University Press, 1999 (paperback 2001).
We will cover Chapter 7 (including Appendix 1).

Contents of Chapter 7:

  1. Symmetric functions in general
  2. Partitions and their orderings
  3. Monomial symmetric functions
  4. Elementary symmetric functions
  5. Complete homogeneous symmetric functions
  6. An involution
  7. Power sum symmetric functions
  8. Specializations
  9. A scalar product
  10. The combinatorial definition of Schur functions
  11. The RSK-algorithm
  12. Some consequences of the RSK-algorithm
  13. Symmetry of the RSK-algorithm
  14. The dual RSK-algorithm
  15. The classical definition of Schur functions
  16. The Jacobi-Trudi identity
  17. The Murnaghan-Nakayama rule
  18. The characters of the symmetric group
  19. Quasisymmetric functions
  20. Plane partitions and the RSK-algorithm
  21. Plane partitions with bounded part size
  22. Reverse plane partitions and the Hillman-Grassl correspondence
  23. Applications to permutation enumeration
  24. Enumeration under group action
Contents of Appendix 1:
  1. Knuth equivalence and Greene's theorem
  2. Jeu de Taquin
  3. The Littlewood-Richardson Rule

Reference texts:
[Fu] W. Fulton, Young tableaux , Cambridge University Press, 1997.
[La] A. Lascoux, Symmetric functions and combinatorial operators on polynomials, AMS, 2003.
[Ma] I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd edition, Oxford University Press, 1995 (paperback 1999).
[Sa] B. E. Sagan, The symmetric group, 2nd edition, Springer-Verlag, 2001.
[EC1]   R. P. Stanley, Enumerative combinatorics, vol. 1, 2nd edition, Cambridge University Press, 2011/2012.