Course Description
and Information 


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 12:302:30pm 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.
Announcements: (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 rightclick (controlclick) 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 ("lefttoright" and "righttoleft"). By default, Sage uses "lefttoright"
but there is a parameter that you can set so that it uses "righttoleft" 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 noncommutative 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.


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class day =



midterm/final (tentative) =



no class =



May 5


The Symmetric group, permutations, cycle
structure, matrix representations, Gmodules, reducibility S1.11.3

May 7


Complete reducibility and Maschke's theorem S1.41.5
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 1.71.9

May 26


Decomposition of the group algebra, Restriction, induction S 1.91.12

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 : SpringerVerlag, 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, c19971999.
 Zelevinsky, A. V. Representations of finite classical
groups : a Hopf algebra approach, Berlin ; New York : SpringerVerlag, 1981.
Algebra
 Sweedler, M. E., Hopf algebras, New York : W.A.
Benjamin, 1969.


