Combinatorial Theory

Class meets: Tuesday and Thursday, 2:30-4 pm, room 12-142

Instructor: Alexander Postnikov

Office hours: TR 4-5pm in Room 2-389.

Grader: Pavlo Pylyavskyy

Description for the Catalog:
Topics for Fall 2006 include enumerative and algebraic combinatorics related to representations of the symmetric group, symmetric functions, and Young tableaux.

Synopsis:
This course is an introduction to algebraic combinatorics that came from representation theory of the symmetric group. The class will cover the classical topics such as Specht modules, Young symmetrizers, Young tableaux, the branching rule, Gelfand-Tzetlin bases, Schur functions, the Robinson-Schensted-Knuth correspondence, the Jacobi-Trudi identity, the hook-length formula, the Littlewood-Richardson rule, the Murnaghan-Nakayama rule, Schützenberger's involution and jeu de taquin, etc. If time allows, some of the following more recent topics will be included: Hecke algebras, Okounkov-Vershik construction, Jucys-Murphy elements, Schur positivity, connections with Horn's Problem and with tropical geometry, Macdonald polynomials, noncommutative symmetric functions, etc.

The class should be accessible for first year graduate students. The prerequisites are minimal.

Course Level: Graduate

Recommended textbook:
*  Bruce E. Sagan: The Symmetric Group, 2nd Edition, Graduate Texts in Mathematics, Springer, 2001.

Additional reading:
*  Richard P. Stanley: Enumerative Combinatorics, Volumes 1 and 2, Cambridge University Press, 1996 and 1999.
*  I. G. Macdonald: Symmetric Functions and Hall polynomials, 2nd Edition, Clarendon Press, Oxford, 1995.
*  William Fulton: Young Tableaux, Cambridge University Press, 1997.
*  William Fulton, Joe Harris: Representation Theory, A First Course, Graduate Texts in Mathematics, Springer, 1991.
*  Gordon James, Adalbert Kerber: The Representation Theory of the Symmetric Group, Cambridge University Press, 1984.

Grading: Based on several problem sets

Problem Sets:

• Problem Set 1 (due September 21)
• Problem Set 2 (due October 12)
• Problem Set 3 (due October 26)
• Problem Set 4 (due November 16)
• Problem Set 5 (due December 12)

Lectures:

• Lecture Notes: Lectures 1-6   Lectures 7-9   Lectures 10-17
  I thank the notetaker (who wishes to remain anonymous) and Alexey Spiridonov (for scanning the notes).

1. R 09/07/2006. Introduction.

2. T 09/12/2006. Representations of S_n and the Young lattice. Young tableaux as paths. Up and down operators.

   Representation theory basics. Group algebra. Maschke's theorem.

3. R 09/14/2006. Schur's lemma and its corollaries. Characters. Conjugacy classes. Orthogonality of characters.

4. T 09/19/2006. Characters (cont'd). Character table for S_4. Wiring diagrams and Coxeter relations.

   Okounkov-Vershik construction: Branching graph and Gelfand-Tsetlin bases.

   For more details, see:
   • A. Vershik, A. Okounkov: A new approach to the representation theory of the symmetric groups, II. math.RT/0503040
   • A. Vershik: A new approach to the representation theory of the symmetric groups, III: Induced representations and the Frobenius--Young correspondence. math.RT/0609258
   • P. Py: On representation theory of symmetric groups. PDF

5. R 09/21/2006. Okounkov-Vershik construction (cont'd). The center of the group algebra. Gelfand-Tsetlin subalgebra. Jucys-Murphy elements.

6. T 09/26/2006. Okounkov-Vershik construction (cont'd). Example: Standard (n-1)-dimensional representation of S_n. Jucys-Murphy elements satisfy degenerate Hecke algebra relations. Content vectors.

7. R 09/28/2006. Okounkov-Vershik construction (end).

   Hooklength formula and hook-walks.

   For more details, see:
   • C. Greene, A. Nijenhuis, H. S. Wilf, A probabilistic proof of a formula for the number of Young tableaux of a given shape. Adv. in Math. 31 (1979), no. 1, 104--109.

8. T 10/03/2006. Schur functions via semi-standard tableaux. Cauchy identity. Robinson-Schensted-Knuth (RSK) correspondence.

9. R 10/05/2006. Properties of RSK. Fomin's growth diagrams.

   T 10/10/2006. Columbus Day Vacation

10. R 10/12/2006. RSK (cont'd). Longest increasing subsequences. Gelfand-Tsetlin patterns.

11. T 10/17/2006. q-binomials coefficients. Unimodality of Gaussian coefficients.

12. R 10/19/2006. Problem set review.

13. T 10/24/2006. The ring of symmetric functions and its various bases.

14. R 10/26/2006. Lindstrom's lemma and Jacobi-Trudi identity. The classical definition of Schur polynomials. The determinantal formula for the number of SYT's.

15. T 10/31/2006. q-analogue of the determinantal formula and of the hook-length formula. Reverse plane partitions. Hillman-Grassl correspondence.

16. R 11/02/2006. Problem set review.

17. T 11/07/2006. Plane partitions, noncrossing paths, rhombus tilings, perfect matchings, and pseudoline arrangements. Viennot's shadow construction for RSK.

18. R 11/09/2006. Green's theorem. P-equivalence and Knuth's equivalence. Schutzenberger's jeu de taquin.

19. T 11/14/2006. The Littlewood-Richardson rule via jeu de taquin.

20. R 11/16/2006. The classical LR-rule (via lattice words).

21. T 11/21/2006. Variants of the LR-rule: Zelevinsky's pictures, Berenstein-Zelevinsky's triangles, Knutson-Tao's honeycombs and puzzles. Symmetries of the LR-coefficients.

    R 11/23/2006. Thanksgiving Vacation

22. T 11/28/2006. The Murnaghan-Nakayama rule. The Frobenius characteristic map. The characters of the symmetric group.

23. R 11/30/2006. Problem Set review.

    For more details on Problems 7 and 8(b) (noncommutative Schur functions, the Edelman-Greene correspondence, and related stuff), see the following papers:

    • S. Fomin, C. Greene: Noncommutative Schur functions and their applications, Discrete Mathematics 193 (1998), 179-200.
    • R. Stanley: On the number of reduced decompositions of elements of Coxeter groups,” European. J. Combin. 5 (1984), 359–372.
    • P. Edelman, C. Greene: Balanced tableaux, Adv. Math. 63 (1987), 42-99.

24. T 12/05/2006. Guest lecture by Richard Stanley.