The course will introduce you to representation theory, as viewed "from infinity". For example, along with representations of a single symmetric group $S(n)$ for fixed $n$, we will be interested in the asymptotic behavior of the theory as $n\to+\infty$. A significant part of this asymptotics is captured by the theory of characters of the infinite symmetric group $S(\infty)$, which is the inductive limit of the $S(n)$'s. The asymptotic point of view requires new tools, as many of the existing representation-theoretic approaches break down. The description of the representation theory of $S(\infty)$ is at the same time easier and more complicated than that of the individual $S(n)$'s. Many of the answers here are given in probabilistic language.
This course is a beginner's introduction to the subject. We will discuss symmetric functions, probabilistic exchangeability, characters and representations of finite and infinite symmetric groups, and limit shapes of random partitions. The first half of the course will correspond to the first part of the textbook by Borodin-Olshanski. The second half will discuss probabilistic / math-physics applications of the theory. Necessay prerequisies include some algebra background (preferrably at graduate level), some probability background (1 undergraduate course is fine), and basic understanding of analysis (measures, Lebesgue integral --- first month of UVA's graduate real analysis suffices).
(60%) weekly homework (posted to this website, see below) with 2.5- or 3-week deadlines
(40%) take-home final exam with a 1-week deadline
Level of homework problems ranges from easy to very difficult. It is understood that you will not turn in all problems all the time, but putting an adequate effort into solving homework problems and communicating your solutions clearly is of paramount importance for your learning.
Each homework is a "pass" if you correctly solve at least 2 problems. To get full credit for the homework, you need 7 "passes". I accept homework problems in hard copies or by email (firstname.lastname@example.org)
After my grading of the homework, you will have 1 week to resubmit corrections if you'd like, improving your chances of a pass.
The final exam will consist of several problems similar to easier of the homework problems.
Each homework assignment will have due date. Late assignments are not accepted. If you have special needs, emergency, or unavoidable conflicts, please let me know as soon as possible, so we can arrange a workaround.
All students with special needs requiring accommodations should present the
appropriate paperwork from the Student Disability Access Center (SDAC). It is
the student's responsibility to present this paperwork in a timely fashion and
follow up with the instructor about the accommodations being offered.
Accommodations for midterms or final exams (e.g., extended time) should be arranged at
least 5 days before an exam.
The University of Virginia Honor Code applies to this
class and is taken seriously.
Any honor code violations
will be referred to the
Collaboration on homework assignments
Group work on homework problems is allowed and strongly encouraged. Discussions are in general very helpful and inspiring. However, before talking to others, get well started on the problems, and contribute your fair share to the process. When completing the written homework assignments, everyone must write up his or her own solutions in their own words, and cite any reference (other than the textbook and class notes) that you use. Quotations and citations are part of the Honor Code for both UVa and the whole academic community.
I am very grateful to Jeanne Scott for numerous helpful discussions of the Young-Fibonacci graph and its boundary. The author is supported by the NSF grant DMS-2153869 and the Simons Collaboration Grant for Mathematicians 709055.
"The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions" by B. Sagan (1991) - introduces basic representation theory of finite groups, develops representations of the $S(n)$'s, treat its combinatorial aspects, discusses symmetric functions