Math 669 Winter 2024
Symmetric Functions

Lectures: Tuesday and Thursday 1-2:30pm 4088 East Hall

Instructor: Thomas Lam,

Office Hours: After lectures.

This is a graduate level mathematics course and intended to be challenging. Students are expected to have taken algebra at the level of 593-594, and have had previous experience with combinatorics at the graduate level. Concepts from geometry and topology may be used.

Grading: There will be problem sets roughly once a month. Students will be asked to write a term paper, roughly 5 pages in length. There will be no exams.
Grades will be calculated from: Paper (25%) and Problem Sets (75%).

References: We will not follow a textbook closely.
Enumerative Combinatorics Vol. 2, Richard Stanley
Young tableaux, William Fulton
Symmetric Functions and Hall Polynomials, Ian Macdonald

Pset problems: Pset problems (will be updated throughout the semester. Submit five problems on or before each of the following dates: Feb 22, Mar 14, Apr 11.
Note that the problems range from standard exercises to tricky or difficult problems! Each problem, regardless of difficulty, is worth 5% of your final grade. Correct solutions to any fifteen problems, and a good final paper, would give a grade of A. Correct solutions to difficult problems will in addition gain my respect.

Homework must be written in LaTeX.
You are encouraged to work with other students on the problem sets, but you must include the names of those you worked with when you hand in your homework. You are not allowed to post homework problems on question websites such as mathoverflow or stackexchange. If you use a solution you find in a book, online, or elsewhere, you must acknowledge the source.

Academic Misconduct: The University of Michigan community functions best when its members treat one another with honesty, fairness, respect, and trust. The college promotes the assumption of personal responsibility and integrity, and prohibits all forms of academic dishonesty and misconduct. All cases of academic misconduct will be referred to the LSA Office of the Assistant Dean for Undergraduate Education. Being found responsible for academic misconduct will usually result in a grade sanction, in addition to any sanction from the college. For more information, including examples of behaviors that are considered academic misconduct and potential sanctions, please see

LSA is committed to delivering our mission while aiming to protect the health and safety of the community, which includes minimizing the spread of COVID-19. Our entire LSA community is responsible for protecting the collective health of all members by being mindful and respectful in carrying out the guidelines laid out in our Wolverine Culture of Care and the University’s Face Covering Policy for COVID-19. Individuals seeking to request an accommodation related to the face covering requirement under the Americans with Disabilities Act should contact the Office for Institutional Equity.
In our classrooms all students are expected to adhere to the required safety measures and guidelines of the State of Michigan and the University of Michigan, wearing a face covering that covers the mouth and nose in all classrooms, and not coming to class when ill or in quarantine. It is important to also be thoughtful about group gatherings as well as about classroom activities and exercises that require collaboration.
Any student who is not able and willing to comply with campus safety measures for this course should contact the course instructor or their academic advisor to discuss alternate participation or course options. Students who do not adhere to these safety measures while in a face-to-face class setting, and do not have an approved exception or accommodation, may be asked to disenroll from the class.
For additional information refer to the LSA Student Commitment to the Wolverine Culture of Care and the OSCR Addendum to the Statement of Student Rights and Responsibilities on the OSCR website.

Disabilities: The University of Michigan recognizes disability as an integral part of diversity and is committed to creating an inclusive and equitable educational environment for students with disabilities. Students who are experiencing a disability-related barrier should contact Services for Students with Disabilities; 734-763-3000 or For students who are connected with SSD, accommodation requests can be made in Accommodate. If you have any questions or concerns please contact your SSD Coordinator or visit SSD’s Current Student webpage. SSD considers aspects of the course design, course learning objects and the individual academic and course barriers experienced by the student. Further conversation with SSD, instructors, and the student may be warranted to ensure an accessible course experience.

Course Recordings: Course lectures audio/video will be recorded. Please be aware that as part of your participation in this course, you may be recorded. If you have concerns about this, please let me know as soon as possible.

Topics covered (tentative):
Bases of symmetric functions, Schur functions, Young tableaux, Robinson-Schensted algorithm, symmetric group characters, characters of GL(n), Littlewood-Richardson rule, quasisymmetric functions, plactic monoid, Stanley symmetric functions

List of lectures: