Representation theory on the symmetric group is one of the most fundmental concepts in mathematics. We will cover the basic notions in representation theory such as characters, irreducible representations, modules, and induced representations; all with the symmetric group as the prime example. The irrecible modules will be constructed using a combinatorical notion of tableaux. we will state and prove the Murnaghan–Nakayama algorithm for computing the character table of the symmetric group on a finite set. Then we will describe some of the close connections between representations of symmetric groups and symmetric functions. So much is known about the symmetric group, and yet there are still questions on the cutting edge of research.
Prerequisits: This is a topics course aimed at advanced undergraduates who are comfortable with reading and writing proofs. Math 300 and some coursework in linear algebra is required. Math 402 would be helpful, but we will review the concept of a group and the basics of representation theory.
Grading: This class will not be graded in the standard format. Instead, students will work with the instructor to set goals and measure their individual success. Contributions in the form of homework, quizes, solutions to bonus problems, expository writing related to the course topics, and attendance in seminars can be counted toward the student's portfolio.
Tests: Representation theory is a beautiful subject, but it takes hard work to become a master at it. Listening to someone else explain a solution is not the same as coming up with it yourself. We will have two quizzes and a final exam in this class so you can show what you know as an individual. The tests will be based on material covered in the lectures, plus the recommended reading assignments. Anything covered in class is fair game whether it appears in the textbook or not. You are responsible for taking notes so you have a complete record of what we discuss. Also, be sure to fill in the gaps if you don't fully understand something. I am always happy to clarify questions before or after class.
Discussion Time: We will have a room reserved every Friday 1:30-2:30. This space is open to everyone in the class to get together to work on homework. I will hold office hours at the same time.