Symmetric functions are ubiquitous in many areas of mathematics. They arise naturally in group theory when studying the characters of the symmetric group and the general linear group. They also appear in the study of the geometry of grassmannians and flag varieties, and are an important tool in algebraic combinatorics and representation theory.
The main aim of the course will be to provide a solid introduction to the basics of symmetric polynomials and symmetric functions. After covering these basics, we will discuss applications of symmetric functions to other areas of mathematics. In particular, we will cover the following topics.
- Partitions and Young tableaux
- Symmetric polynomials and the ring of symmetric functions
- Elementary and complete homogeneous symmetric functions
- Power sum symmetric functions
- Schur functions and orthogonality
- Hopf algebra structure on the ring of symmetric functions
- Characters of the symmetric group
- Bosonic Fock space and the boson-fermion correspondence
Student should have completed introductory courses in group theory, and the theory of rings and modules. For example, students having completed uOttawa courses MAT 3143/3543 will have the required prerequisites. Completion of MAT 4141/5141 is an asset, but not a requirement.
Click the above header to download lecture notes for the course. They are a single file that may be updated as the course progresses. These notes are the main reference for the course.
There are no required textbooks for this course. However, students looking for additional references may find the following useful.
- Symmetric functions and Hall polynomials, I. G. Macdonald
- An introduction to symmetric functions and their combinatorics, E. S. Egge
- Introduction to symmetric functions, M. Zabrocki
There are two main locations for information regarding the course:
- alistairsavage.ca/symfunc/. This is the main course website. It will be updated regularly and contains important material for the course.
- Brightspace. Assignments will be submitted and returned through Brightspace. Grades will also be posted on Brightspace.
The language of instruction for this course is English. However, all students have the right to produce their written work and to answer examination questions in either English or French.