Math 118: Combinatorial Representation Theory

Spring 2018

Syllabus



Course Description

This course in an introduction to combinatorial representation theory and related topics such as symmetric functions. My goal is to discuss the following topics:
  • An overview of representations of finite groups and finite dimensional algebras
  • There will be an emphasis on the representation theory of the symmetric group, including Young symmetrizers, specht modules, branching rules, and Gelfand-Tzetling bases.
  • Polynomial representation theory of the general linear group.
  • Symmetric functions and how it is connected to representation theory
  • Schur-Weyl Duality
  • Young tableaux combinatorics, RSK, hook length formula
  • Partition algebra and connections to representation theory of the symmetric group
This is only the plan! I hope to stick to it for the most part. However, I reserve the right to deviate from this plan. In choosing the topics, I took into consideration the things that I find beautiful. As G.H. Hardy put it in his book "A Mathematician's Apology",

    "The mathematician's patters, like the painter's or the poet's, must be beautiful; the ideas, like the colors or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics."

The objectives of the course are:

  • Students will learn basics of combinatorial representation theory.
  • Students will be able to solve problems related to the topics discussed in class.
  • Students will practice their oral communications and presentation skills overall.
Grades
The course grade will be computed as follows:

Percent of Final Grade
Homework 50
Participation 20
Class Presentation 30

Students will be graded on class participation. This means that you should come to class, ask questions and contribute ideas during class.
Class Presentation
The final project is to read a current research article related to the topics discussed in class and give a 20 minute talk about it similar to the AMS special session talks and provide constructive criticism on other students presentations.
Homework Policy
Written homework will be assigned roughly every other week.
  • Unexcused late and missing papers count zero.
  • Homework is to be written using latex whenever possible.
  • All papers are to be stapled.
  • Please mention on your problem set the names of the students that you worked with, and also reference any articles, books or websites if your solution takes significant ideas from them.
Textbook
There is no textbook for this course as we will be jumping around between different sources. Here are a few good references for the topics that we will be discussing in this class:
  • W. Fulton and Joe Harris, Representation theory a first course
  • B. Sagan, The symmetric group
  • R. Stanley, Enumerative Combinatorics, Vol. II, Chapter 7.
  • I. MacDonald, Symmetric Functions
  • W. Fulton, Young Tableaux.
    • T. Ceccherini-Silberstain, F. Scarabotti and F. Tolli, Representation Theory of the symmetric group.
Honor Principle
Students are encouraged to work together to do homework problems. What is important is a student's eventual understanding of homework problems, and not how that is achieved.

The honor principle on homework: What a student turns in as a homework solution is to be his or her own understanding of how to do the problem. Students must state what sources they have consulted, with whom they have collaborated, and from whom they have received help. It is a violation of the honor code to copy solutions from problems posted on the web or book or any other source. The solutions you submit must be written by you alone. Any copying (electronic or otherwise) of another person's solutions, in whole or in part, is a violation of the Honor Code. For example, it is a breach of the honor code to read the solutions of someone else in order to write your solution.

If you have any questions as to whether some action would be acceptable under the Academic Honor Code, please speak to me I will be glad to help clarify things. It is always easier to ask beforehand than to have trouble later!
Disabilities and Religious Observances
Students with disabilities enrolled in this course and who may need disability-related classroom accommodations are encouraged to make an appointment to see your instructor before the end of the second week of the term. All discussions will remain confidential, although the Student Accessibility Services office may be consulted to discuss appropriate implementation of any accommodation requested.

Some students may wish to take part in religious observances that occur during this academic term. If you have a religious observance that conflicts with your participation in the course, please meet with your instructor before the end of the second week of the term to discuss appropriate accommodations.

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