Math 155r: Algebraic Combinatorics -- Spring 2019

Lectures: Mondays and Wednesdays 1:30-2:45pm, Science Center 507.

Professor: L. Williams (Science Center 510, e-mail williams@math.harvard.edu)

Office Hours: Mondays 3-4pm and by appointment.

Course assistant: Alec Sun, e-mail suna@college.harvard.edu.

Alec's section: Fridays 1:30-2:30pm in Science Center 222.

Alec's office hours: Fridays 2:30pm-4pm in the 4th floor Math Lounge.

Course description

This course is an introduction to algebraic combinatorics that comes from the representation theory of the symmetric group. We will start with a quick overview of the representation theory of finite groups, and then cover topics such as Young tableaux and Young symmetrizers, Specht modules, Jucys-Murphys elements, the hook-length formula, the branching rule, Gelfand-Tsetlin bases, Schur functions, the Littlewood-Richardson rule, the Robinson-Schensted-Knuth correspondence, Schutzenberger's involution and jeu de taquin, characters of the general linear group and Schur-Weyl duality.

Prerequisites: linear algebra and abstract algebra (such as Math 122). Familiarity with representation theory of finite groups would be helpful but is not required.

Grades will be based on problem sets (50%), an exam (15%), and a final paper (35%).

References

A main reference for the course will be Bruce Sagan's book ``The symmetric group." I will also draw from Fulton and Harris' ``Representation theory," Fulton's ``Young tableaux," and Stanley's ``Enumerative Combinatorics 2."

Lectures

  • Lecture 1 (Jan. 28): Introduction + start review of representation theory of finite groups
  • Lecture 2 (Jan. 30): More representation theory and examples
  • Lecture 3 (Feb. 4): Characters of representations
  • Lecture 4 (Feb. 6): Characters of representations; start the RSK correspondence.
  • Lecture 5 (Feb. 11): RSK correspondence and Schensted's theorem.
  • Lecture 6 (Feb. 13): RSK: Schutzenberger's theorem and Viennot's shadow line construction.
  • President's day (Feb 18), no lecture
  • Lecture 7 (Feb. 20): Specht modules.
  • Lecture 8 (Feb. 25): Finish Specht modules.
  • Lecture 9 (Feb. 27): Vershik/Okounkov approach to representation theory of symmetric group.
  • Lecture 10 (Mar. 4): Vershik/Okounkov approach (continued).
  • Lecture 11 (Mar. 6): The hook-length formula.
  • Lecture 12 (Mar. 11): Symmetric polynomials.
  • Lecture 13 (Mar. 13): Bases of symmetric polynomials (monomial, elementary)
  • Spring break (Mar. 18-22): no lecture
  • Lecture 14 (Mar. 25): Review for midterm
  • MIDTERM (Mar. 27)
  • Lecture 15 (Apr. 1): Complete homogeneous symmetric functions; start Schur functions.
  • Lecture 16 (Apr. 3): Schur functions and semistandard tableaux.
  • Lecture 17 (Apr. 8): RSK for semistandard tableaux; Cauchy identity.
  • Lecture 18 (Apr. 10): Symmetry of skew Schur functions; a scalar product.
  • Lecture 19 (Apr. 15): Guest lecture by Bruce Sagan (chromatic polynomial).
  • Lecture 20 (Apr. 17): The characteristic map from class functions to symmetric functions.
  • Lecture 21 (Apr. 22): Knuth equivalence, Green's Theorem, jeu de taquin.
  • Lecture 22 (Apr. 24): Recap of the main ideas of the class!!
  • Lecture 23 (Apr. 29): Representations of the general linear group.
  • Lecture 24 (May 1): Grassmannians and flag varieties.
  • (May 8): FINAL PROJECT DUE (submit to canvas).