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. Please feel free to unmute yourself and ask a question or make a comment at any moment during zoom sessions.

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.

Problem Sets:

problem set 1  due friday, october 9, 2020

problem set 2  due friday, december 4, 2020

Submit your solutions on canvas


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. RSK (cont'd). Insertion algorithm. Symmetry of RSK. Increasing and descreasing subsequences. Application: Erdos-Szekeres theorem. Greene's theorem. Young's lattice. Notes

  8. M 09/21/2020. Up and Down operators. Differential posets. Examples: Young's and Fibonacci lattices. Oscillating tableaux. Generalized Schensted correspondence: bijection between oscillating tableaux and rook placements. Notes

  9. W 09/23/2020. Fomin's growth diagrams. Notes

  10. F 09/25/2020. Piecewise-linear RSK. Semi-standard growth diagrams. Toggles. Notes

  11. M 09/28/2020. More on toggles. Bender-Knuth involutions. RSK vs Hillman-Grassl correspondence. The hook length formula. Stanley's hook-content formula. Weyl's dimension formula. Notes

  12. W 09/30/2020. Gelfand-Tsetlin polytopes. Ehrhart polynomials. Kostant's partition function. Kostant's weight formula. Flow polytopes. Ehrhart positivity conjecture. Klyachko's saturation conjecture. Notes

  13. F 10/02/2020. Jacobi-Trudi identities. Gessel-Viennot-Lindstrom lemma. Notes

  14. M 10/05/2020. Applications of Jacobi-Trudi identities. The determinantal formulas for #SYT's vs the hook length formula. The exponential specialization and principal specializations of symmetric functions. q-analogs: q-factorial and q-binomial coefficients. Notes

  15. W 10/07/2020. Proof of polynomiality of q-binomial coefficients via Schubert decomposition of the Grassmannian. Symmetry and unimodality of the Gaussian coefficients. Sylvester's proof of the unimodality. Weighted up and down operators. Notes

  16. F 10/09/2020. Problem Set 1 is due: Submit your solutions
    Plane partitions. Counting using Lindstrom's lemma. MacMahon's formula. RPP's vs SSYT's. Rhombus tilings, pseudoline arrangements, and perfect matchings. The arctic circle phenomenon. Notes

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

  17. Tuesday 10/13/2020. (Monday schedule of classes) Discussion of Problem Set 1. Your presentations of solutions. (Volunteers are welcome to make short 5-10 min presentations during the class time. Please let me know in advance if you want to present a solution.)

  18. W 10/14/2020. Introduction to representation theory. Irreducible representations of symmetric groups. Branching rule. Group algebra. Young symmetrizer. Notes

  19. F 10/16/2020. Vershik-Okounkov's "new approach" to the representation theory of the symmetric groups. The center of the group algebra. Bratteli diagram. Gelfand-Tsetlin bases. Gelfand-Tsetlin subalgebra of the group algebra. Notes

  20. M 10/19/2020. Vershik-Okounkov's approach (cont'd). Centers and centralizers of group algebras. Jucys-Murphy elements. The Gelfand-Tsetlin basis = the basis of common eigenvectors of JM-elements. The degenerate affine Hecke algebra (DAHA). Notes

    A. M. Vershik, A. Yu. Okounkov. A New Approach to the Representation Theory of the Symmetric Groups. II: arxiv version and journal version

  21. W 10/21/2020. Vershik-Okounkov's approach (cont'd). Local analysis of the spectrum of Jucys-Murphy elements. Abaci and content vectors of SYT's. Young's orthogonal form. Notes

  22. F 10/23/2020. Characters of representations. Character tables for S_3 and S_4. The Murnaghan-Nakayama rule. Ribbon tableaux. Alternating permutations. Notes

  23. M 10/26/2020. Multivariate generalization of the Murnaghan-Nakayama rule and scattering amplitudes in N=4 SYM theory. Parke-Taylor factors. Kleiss-Kuijf relations. Ribbon and tree relations. Notes

    N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, A. Postnikov, J. Trnka. On-shell structures of MHV amplitudes beyond the planar limit, J. High Energy Phys. 2015, no. 6, 179: arXiv version.

  24. W 10/28/2020. Proof of the (generalized) Murnaghan-Nakayama rule. The "game" on graphs. Non-crossing and non-nesting alternating trees and the Catalan numbers. Chan-Robbins-Yuen conjecture (proved by Zielberger). Kostant's partition function. Subdivisions of root polytopes. The Orlik-Terao algebra. The tree, forest, and Parke-Taylor S_n-submodules of the Orlik-Terao algebra. Notes

  25. F 10/30/2020. Frobenius character formuala. Proof of the Murnaghan-Nakayama rule for symmetric functions (using the involution principle). Notes

  26. M 11/02/2020. The Hall inner product on symmetric functions. A formula for skew Schur functions. Schutzenberger's jeu de taquin. Bender-Knuth involutions (again). Jdt-slides vs toggle operations. Induced representations of S_n. The regular representation of S_n as the sum of n-ribbons. Notes

  27. W 11/04/2020. The Littlewood-Richardson rule, and its friends. Zelevinsky pictures. Notes

  28. F 11/06/2020. Stembridge's (cf. Berenstein-Zelevinsky) concise proof of the Littlewood-Richardson rule via Bender-Knuth involutions. Zelevinsky's extension of the LR-rule. LR-rule via Gelfand-Tsetlin patterns. Notes

  29. M 11/09/2020. LR rule (cont'd): Gelfand-Tsetlin patterns, Berenstein-Zelevinsky triangles, and Knutson-Tao honeycombs. Notes

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

  30. F 11/13/2020. LR rule (cont'd): Bijections between Littlwood-Richardson tableaux, Berenstein-Zelevinsky triangles, and Knutson-Tao honeycombs. Puzzles. Notes

  31. M 11/16/2020. Puzzles vs honeycombs. Symmetries of the LR-coefficients. Horn's problem and the Klyachko cone. Notes

  32. W 11/18/2020. Saturation theorem. Berenstein-Zelevinsky polytopes and cones. Description of the Klyachko cone. PRV conjecture. Notes

  33. F 11/20/2020. Description of the Klyachko cone. PRV conjecture. Sums, differences, and x-rays of permutations. The self-dual Hopf algebra of symmetric functions. Notes

    M 11/23/2020. Thanksgiving vacation.

    W 11/25/2020. Thanksgiving vacation.

    F 11/27/2020. Thanksgiving vacation.

  34. M 11/30/2020. More on the hook length formula... The probabilistic "hook walk" proof of Greene, Nijenhuis, Wilf. The shifted hook length formula. Notes

  35. W 12/02/2020. Naruse's "excited" hook length formula for skew shapes. Parking functions. Tree inversion polynomial. Alternating permutations. Notes

  36. F 12/04/2020. Problem Set 2 is due: pdf
    More on alternating permutations & parking functions. Representations of symmetric groups and symmetric functions associated with parking functions. Generalized parking functions. Strips and ribbons. Notes

  37. M 12/07/2020. Generalized parking functions. Symmetric trees. A sign-reversing involution on gamma-parking-function tableaux. Chromatic polynomials and Stanley's chromatic symmetric functions. Acyclic orientations of graphs and the refinement of acyclic orientations by the number of sinks. Notes

  38. W 12/09/2020. Positivity of symmetric functions: e-positivity and Schur positivity. Claw-free graphs and (3+1)-free posets. Stanley's conjecture on e-positivity of chromatic symmetric functions for (3+1)-free posets. Gasharov's Schur-positivity theorem. Unit interval orders and the Catalan numbers. Okounkov's Schur positivity conjecture. LPP theorem and Schur log-concavity. Notes

Recommended textbooks:

Related courses taught in the past:

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: December 9, 2020