MATH 180: The symmetric group, symmetric functions and computer exploration
Winter 2015, UC Davis

Lectures: MWF 2:10-3:00pm in Hickey Gym 290
Office hours: M 4-5pm
CRN 93948
Instructor: Anne Schilling, MSB 3222, phone: 554-2326,
  • Bruce E. Sagan, "The symmetric group, Representations, combinatorial algorithms, and symmetric functions", Springer, second edition, 2001.
  • Symmetric functions in Sage
Other very useful texts:
  • William Fulton, "Young tableaux", London Mathematical Society, Student Texts 35, Cambridge University Text 1997
  • Richard P. Stanley, "Enumerative Combinatorics, Volume II" Cambridge Studies in Advanced Mathematics 62, Cambridge University Press 1999.
  • I.G. Macdonald, "Symmetric functions and Hall polynomials", Oxford Science Publication, second edition, 1995
Prerequisites: MAT 67, MAT 150A; or permission by instructor
Grading: There will be regularly assigned homeworks, which you should present in class on Fridays. Every student should present at least 1 problem in class. The only way to really learn and grasp the material is to play and work with it yourself! There will also be projects to be presented in groups of 2-3 students at the end of the quarter.
Computing: During class, I will illustrate some results using the open source computer algebra system Sage. When you follow the link, you can try it out yourself using SageMathCloud. Or you can sign up for a Class Account with the math department. Log into and type the command `sage` to launch a Sage session in the terminal.

Course description

The representation theory of the symmetric group has beautiful descriptions in terms of combinatorics. We will discuss the Specht modules and combinatorial implications such as the Robinson-Schensted-Knuth algorithm and jeu de taquin. Much of this theory can be recast in the language of symmetric functions. In particular, we will discuss the Schur functions combinatorially in terms of Young tableaux and their interpretation as symmetric group characters. A lot of the material can also be explored computationally.

Lecture topics

Sage Worksheets

Very basic tutorial in Sage
Basic Python and Sage tutorial
Tutorial on group theory in Sage
Tutorial on combinatorics in Sage
Worksheet 1 on the symmetric group
Worksheet 2 on how to compute the character of the defining representation of S_n (see HW2)
Worksheet 3 on dominance order
Worksheet 4 on group homology written by Ruian Chen
Worksheet 5 on symmetric group representations
Decomposition of the defining representation of S_n written by Nicholas Schirtzinger


Homework 1 due January 16
Homework 2 due January 23
Homework 3 due January 30
Homework 4 due February 6
Homework 5 due February 13 (extended to February 18)
Homework 6 due February 27
Homework 7 due March 6
Homework 8 due March 13