Course Description and Information

Course Description

A list of handouts

Math 6161- Algebraic Combinatorics :
Symmetric Functions

    Course Description:

We will learn the representation theory of finite groups, Young's construction of the irreducible representations of the symmetric group and the isomorphism of the ring of representations of the symmetric group with the Hopf algebra of symmetric functions. These become the basic tools for decomposing many symmetric group modules into irreducible components.

We will use Sage to do computations. The programs for doing symmetric function computations are already built into Sage and we need only develop programs for computing characters of modules. This will take some patience and willingness to learn Sage (a language based on python). Since I don't see a better way of learning than doing, after covering the basics of what we need do computations, I will pull out a list of descriptions of symmetric group modules and we will use the programs we develop to compute examples. Our computations will be used to conjecture formulas and I'll show you which modules we can give explicit formulas for with our current techniques, and which ones are still open.

This course meets twice a week, Monday/Wednesday from 12:30-2:30pm in Ross S156.

Part of this time will be lecture/problem session and we will spend part of this time in a computer lab (where depends on the number of students attending the class).

The representation theory component of the course will follow "The Symmetric Group : Representations, Combinatorial Algorithms & Symmetric Functions" by Bruce Sagan plus some additional notes that I will provide.  The part of the course that covers symmetric functions will follow notes that I have written/will be updating that covers symmetric functions from a plethystic notation and Hopf algebra perspective. Other references for symmetric functions are Sagan's book, R. Stanley's "Enumerative Combinatorics Vol 2", and I.G. Macdonald's "Symmetric Functions and Hall Polynomials".

Prof. Mike Zabrocki
Office: TEL 2028
Office Hours: by appointment (Monday and Wednesday)
email :

Course Evaluation:
Midterm: 30%
Homework and Labs: 30%
Final Exam: 40%

Check the schedule for dates of the midterm/homeworks/final. I will be out of town the last week of June/first week of July for a conference. The course is only 4 hours a week (instead of the usual 6) because I find that it is hard to cover and absorb this material in such a short period of time.

(May 14, 2014) It has come to my attention that May 19 is a statutory holiday and the campus will be closed that day. We will have to make up the days at the end of the course that I will be out of town (June 30 and July 2). I am hoping that this will be easier to do once we start doing more computer lab. We need to have roughly 36 hours of class for a 3.0 credit course = 18*(2 hours).

Here is a list of homework problems I have assigned in class. I will add solutions after they are presented in class if typing them up is not too onerous.
Homework 1
Homework 2

(May 21, 2014) I asked for a room for this course with more board space. The registrars office gave us Ross S101A starting TODAY. I sent around an email to some of the people in the class (but potentially not all). If you see people who have been attending, please let them know.

(May 22, 2014) I've asked for another room. We will occupy the seminar room if necessary because that other one is crazy.

(May 26, 2014) We got a room in HNE 030. Cross your fingers that it is a better room.

(May 27, 2014) Here is the handout that I gave yesterday as practice for the midterm.

(June 2, 2014) Pavel Shuldiner has typed up the notes for this course. They will be updated should we find typos and corrections that need to be made. If you notice mistakes, please pass them along to myself or Pavel.

(June 2, 2014) Here is a copy of the exam from Wednesday. I thought it went really well and we will discuss it in class today. I will use it as a starting point for learning sage.

(June 2, 2014) Here is the worksheet from the sage cloud server. Use this version if you sign into the same sever that I did. You may need to right-click (control-click) and save as a file if you want to save it to your own computer and upload it. If you are using sage on your own computer you will want to use this worksheet instead. If all else fails, I am also providing a pdf of the worksheet with the text.

(June 4, 2014) Here is a set of 4 problems that we can work on in class today. Bring your laptop or share with someone who has one.

(June 4, 2014) Here is the sage worksheet (using the notebook format and not the sage cloud server). I am also posting the pdf file in case you need to copy into another format. I am going to be using the notebook format because I can run it locally on my computer.

(June 9, 2014) What we are going to do today is construct the irreducible representations of the symmetric group and Young's idempotents in the group algebra. This was a big accomplishment of mathematics in the early 20th century. Young was one of the first mathematicians to provide such a construction, but there were others (e.g. Sagan's presentation shows the work of Specht and Garnir). I am going to follow the presentation that A. Garsia (my advisor) wrote for his class. I like this presentation because (a) I am familiar with his style and (b) he adds lots of details and opinions that you are unlikely to see elsewhere (c) it is extremely algorithmic and easily implemented in a computer language. Another nice thing about this presentation is that it states clearly where it differs from Young's original presentation. We will do this in sage and the programs are in worksheet form here and pdf here.

(June 9, 2014) A few of the identities in the writeup didn't check out once we programed them in. The reason was because there are two multiplications of permutations that we could use ("left-to-right" and "right-to-left"). By default, Sage uses "left-to-right" but there is a parameter that you can set so that it uses "right-to-left" instead. See the note that is in the documentation for SymmetricGroupAlgebra. The worksheets that I posted earlier today are very close to what I did in class, but not exact. If you want to see what I did in class today the files in worksheet form here and pdf here.

(June 10, 2014) I am sick. I was having a hard time in class on Monday and I have since lost my voice. I am going to have to cancel class tomorrow because I don't expect it to come back before then.

Please continue to work on the programming exercises and email me your solutions. If you need help, a hint, or a command please let me know by email.

Also, continue to read through the development of the irreducible representations of the symmetric group that I gave you (and posted on the website). Verify the identities hold when you can through examples.

(June 16, 2014) Here are two homework problems. Do the next one for class June 18. When you finish the second one, send me your Sage functions. Make sure to document your functions.

(June 26, 2014) I posted the worksheet that we did yesterday in class. Please try to do the replacement homework problem (Question #3) that I gave in its place and send me your solutions.

I started to explain the questions for the final exam and I will post more about this later (hopefully today or tomorrow). I will give you a module and I would like you to write programs which will compute the character and then the Frobenius image. I realized that there are a few more things that I need to cover in order for you to be able to finish this problem in a reasonable amount of time. Unfortunately I am going to be out of town next week.

To finish this class we will need to meet July 7 and July 9.

(June 28, 2014) I posted some notes on the final project. It will be to write programs to compute the character of a symmetric group module and its Frobenius image. I only got to the three examples which are submodules of the polynomial ring. I will edit this eventually to include some submodules of the non-commutative polynomial ring.

I think that you know enough mathematics to complete the project, but there is more to learn about how to represent this on the computer. Over the next two weeks I will also post a Sage example so you can see how to do a similar computation on the computer.

(July 7, 2014) I posted the worksheet in pdf and worksheet format from what I did in class today. On Wednesday I will try to finish the programs and I expect you to model your functions after mine to be able to tell me how it decomposes into irreducibles.

(July 9, 2014) Its late, but I posted the worksheet in pdf and worksheet format from what I did in class today. I promised an updated version of the final project, but I am afraid that is going to have to wait until tomorrow.

(July 10, 2014) I updated the descriptions of the the final project. Sam and Farid should do number 4 and 5 respectively. I will email you separately with some instructions.

S Su M Tu W Th F
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10 11 12 13 14 15 16
17 18 19 20 21 22 23
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31 1 2 3 4 5 6
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class day =

midterm/final (tentative) =

no class =

May 5   

The Symmetric group, permutations, cycle structure, matrix representations, G-modules, reducibility S1.1-1.3
May 7

Complete reducibility and Maschke's theorem S1.4-1.5 HW 1
May 12

Example with Machke's Theorem, change of basis, Schur's Lemma S 1.6
May 14

${\mathbb C}[x_1, x_2]$, HW solutions, quotient modules, direct sum, tensor, restriction, HW 2
May 21

Commutant algebra, Group Characters S 1.7-1.9
May 26

Decomposition of the group algebra, Restriction, induction S 1.9-1.12
May 28

Mid term
June 2

generating functions, intro to sage
June 4

Sage to compute characters - HW 3
June 9

irreducible representations of the symmetric group, sage
June 11

no class - professor sick
June 16

more Young's construction of the irreducibles of the symmetric group, RSK - HW 4
June 18

June 23

graded character of ${\mathbb Q}[x_1, x_2, \ldots, x_n]$, induction/restriction, symmetric functions, Frobenius image
June 25

Sage for computing in the group algebra, symmetric functions power, homogeneous and elementary bases, statement of the first three final projects - HW 4 (updated)
July 7

Schur functions, sage and final projects
July 9

sage and final projects

Representation Theory
  • James, G. D., The representation theory of the symmetric groups, Berlin ; New York : Springer-Verlag, 1978.
  • Sagan, B. E., The Symmetric Group : Representations, Combinatorial Algorithms & Symmetric Functions, 1991 and 2001.

Symmetric Functions
  • Macdonald, I. G., Symmetric Functions and Hall Polynomials, Oxford Mathmatical Monographs, Oxford, 1995.
  • Stanley, R., Enumerative Combinatorics, Vol 2., Cambridge ; New York : Cambridge University Press, c1997-1999.
  • Zelevinsky, A. V.  Representations of finite classical groups : a Hopf algebra approach, Berlin ; New York : Springer-Verlag, 1981. 


  • Sweedler, M. E.,  Hopf algebras, New York : W.A. Benjamin, 1969.