MATH 8852: Asymptotic Representation Theory
Email address for communication:
leniapetrov+art2022@gmail.com
Abstract
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 representationtheoretic 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 BorodinOlshanski. The second half will discuss probabilistic / mathphysics 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).
Syllabus items
Office hours
Assessment
There are two components in the course grade:
 (60%) weekly homework (posted to this website, see below) with 2.5 or 3week deadlines
 (40%) takehome final exam with a 1week 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 (leniapetrov+art2022@gmail.com)
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.
Policies
Late homework
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.
Approved accommodations
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.
Honor code
The University of Virginia Honor Code applies to this
class and is taken seriously.
Any honor code violations
will be referred to the
Honor Committee.
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.
Lecture notes

Lecture 1, 8/23
1. Basic representation theory
 1.1 Definitions
 1.2 Irreducible representations
 1.3 Schur's lemma, unitarity, and complete reducibility
 1.4 Regular representation. Irreps of $S(n)$ (formulation)
 1.5 Two examples with asymptotics

Lecture 2, 8/25
 1.6 Characters. The fundamental theorem, and the structure of the regular representation
 1.7 Fourier transforms
 1.8 Positive definiteness and the space $\Upsilon(G)$

Lecture 3, 8/31
 1.8 Positive definiteness and the space $\Upsilon(G)$
 1.9 Irreducible representations, their dimensions and characters of $S(n)$, without proof
2. Character theory of $S(\infty)$. Reduction to branching graphs
 2.1 The group $S(\infty)$ and its conjugacy classes
 2.2 The space $\Upsilon(S(\infty))$ of "characters"
 2.3 Vershik's ergodic theorem (1974) and its consequence for $S(\infty)$
Refs:

Vershik's 1974 paper (in Russian)

OlshanskiVershik (1996) with the proof of the ergodic theorem

Lecture 4, 9/1
 2.4 Restrictions for $S(n)$, without proof. Coherent measures $\{M_k(\lambda)\}$
 2.5 Vershik's ergodic theorem. Example: de Finetti's theorem
 2.6 Branching graphs and the description of $\mathrm{Ex}\, \Upsilon(S(\infty))$

Lecture 5, 9/6
(interlude: SchurWeyl duality from the colloquium)
 2.7 Proof of Vershik's ergodic theorem (if and only if)
3. Branching graphs
 3.1 Branching graphs. General definition, coherent measures, harmonic functions
 3.2 Example: Pascal triangle

Lecture 6, 9/8
 3.2 Example: Pascal triangle (graph, dimension, coherent measures, central measures, harmonic functions)
 3.3 Application of ergodic theorem to branching graphs
 3.4 From extreme characters of $S(\infty)$ to branching graphs
 3.5 The problem of asymptotics of relative dimension

Lecture 7, 9/13
 3.6 Example: Adic shift on Pascal triangle
4. Pascal triangle and its boundary by algebra
 4.1 Pascal triangle and polynomials in two variables: identification with algebra morphisms
 4.2 Relative dimension and boundary of the Pascal triangle

Lecture 8, 9/15
5. Combinatorial properties of the Young graph

5.1 Recursion for dimension, standard Young tableaux (it would be nice to have an associated polynomial algebra)

5.2 Formulas for dimension

5.3 Probabilistic proof of the hook formula

5.4 Another recursion for dimension, operator interpretation

Lecture 9, 9/20
6. Symmetric functions

6.1 The algebra of symmetric functions. Finitedimensional ring, and the inverse limit

6.2 Elementary and complete symmetric functions, their generating functions. Relations

6.3 Power sums, monomial functions (first linear basis). More relations

6.4 The "fundamental theorem" of symmetric functions

Lecture 10, 9/22

6.5 Antisymmetric functions, Schur polynomials, and Schur symmetric functions

6.6 Pieri formula with $p_1=e_1$, and connection to the Young graph

Lecture 11, 9/27
7. Relative dimension in the Young graph

7.1 Determinantal formula for the relative dimension

7.2 Shifted Schur polynomials and relative dimensions

7.3 Shifted symmetric functions $\Lambda^*$

7.4 Modified Frobenius coordinates

Lecture 12, 9/29

7.4 Modified Frobenius coordinates

7.5 Thoma simplex

7.6 Approximation, and proof of Thoma and VershikKerov's theorems

Lecture 13, 10/11
8. Construction of irreducible representations of $S(\infty)$

8.1 Unitary representations and spherical functions

8.2 Motivation: connection to the standard characters as traces in finitedimensional spaces

Lecture 14, 10/13

8.3 Biinvariant functions

8.4 Gelfand pairs and spherical representations

8.5 Gelfand pairs for arbitrary groups

Lecture 15, 10/18

8.6 Realization of irreducible spherical representations of $(S(\infty)\times S(\infty),\mathop{\mathrm{diag}}S(\infty))$
9. qanalogues of Pascal and Young graphs

9.1 Branching graphs with edge multiplicities

Lecture 16, 10/20

9.2 Exchangeability and qexchangeability

9.3 qPascal graph and its boundary

Lecture 17, 10/25

9.4 qPascal graph and infinite Grassmannian over a finite field
10. YoungFibonacci graph

10.1 Differential posets and the commutation relation

10.2 YoungFibonacci graph and encoding of vertices and edges

10.3 Examples of harmonic functions. Plancherel harmonic function

Lecture 18, 10/27

Lecture 19, 11/01
11. Boundary of the YoungFibonacci graph

11.1 Martin boundary

11.2 Type I harmonic functions

11.3 Contraction of harmonic functions to Plancherel

Lecture 20, 11/03

11.3 Contraction of harmonic functions to Plancherel

11.4 Idea of proof of the Martin boundary via clone functions

Lecture 21, 11/10
12. Plancherel growth process

12.1 Plancherel random partitions  reminders

12.2 Plancherel measure and longest increasing subsequences via RSK  historical background and motivation for asymptotics

Lecture 22, 11/17

12.3 Continual Young diagrams  area and hook integral

12.4 VKLS shape as the unique minimizer

Lecture 23, 11/22

12.5 Limit shape of Plancherel measures
13. Hydrodynamics of Plancherel measure and Plancherel growth

13.1 Kerov interlacing coordinates

13.2 Plancherel growth process

13.3 Transition distribution of a continuous Young diagram
Refs:

Mathematica file with the VKLS limit shape

Python code for sampling the (Poissonized) Plancherel Young diagram

VershikKerov 1977 paper (in Russian), the first of their papers on the limit shape

LoganShepp 1977 paper (journal link)

VershikKerov 1985 paper (in Russian), with more detailed estimates of the maximal dimension

A.N. Kirillov (1989), on hook formula and transition probabilities using Kerov interlacing coordinates

Lecture 24, 11/29

13.3 Transition distribution of a continuous Young diagram

Lecture 25, 12/01

13.4 Continuous Plancherel growth process and its limit to the VKLS shape
14. Local correlations in Plancherel measure

14.1 Infinite wedge space computations

Lecture 26, 12/06

14.2 Correlations via determinants, and density

14.2 Correlation kernel via double contour integral

14.3 Asymptotics of density via saddle point method. Another derivation of the VLKS shape
Homework

[posted] Homework 1, due September 13, 2pm

[posted] Homework 2, due September 20, 2pm

[posted] Homework 3, due September 27, 2pm

[posted] Homework 4, due October 11, 2pm

[posted] Homework 5, due October 18, 2pm

[posted] Homework 6, due October 25, 2pm

[posted] Homework 7, due November 3, 2pm

[posted] Homework 8, due November 10, 2pm

[posted] Homework 9, due November 15, 2pm

[posted] Homework 10, due December 9, 2pm

[posted] Homework 11, due December 9, 2pm

[posted] Homework 12, due December 9, 2pm

[posted] Final exam, due December 13, 2pm
Acknowledgements
 I am very grateful to Jeanne Scott for numerous helpful discussions of the YoungFibonacci graph and its boundary. The author is supported by the NSF grant DMS2153869 and the Simons Collaboration Grant for Mathematicians 709055.
Books

"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

"Representations of the infinite symmetric group" by A. Borodin and G. Olshanski (2016)  main textbook for the
initial part of the course

"Asymptotic Representation Theory Of The Symmetric Group And Its Applications In Analysis" by S. Kerov (2003)  a book on asymptotic representation theory

"An Introduction to Symmetric Functions and Their Combinatorics" by E. Egge (2019)  a very gentle introduction to symmetric functions

"Symetric Functions and Hall Polynomials" by I. Macdonald (1995)  excellent reference material on symmetric functions

"Enumerative Combinatorics vol. I" by R. Stanley (1986)  contains material on generating functions, posets, and in particular on differential posets
Papers

A. M. Vershik. Description of invariant measures for the actions of some infinitedimensional groups. Dokl. Akad. Nauk SSSR, 218:4 (1974), 749–752

A. M. Vershik, S. V. Kerov, “Asymptotic theory of characters of the symmetric group”, Funktsional. Anal. i Prilozhen., 15:4 (1981), 15–27; Funct. Anal. Appl., 15:4 (1981), 246–255

Grigori Olshanski, Anatoli Vershik.
"Ergodic unitarily invariant measures on the space of infinite Hermitian matrices".
In: Contemporary Mathematical Physics. F. A. Berezin's memorial volume. Amer. Math. Transl. Ser. 2, vol. 175 (R. L. Dobrushin et al., eds), 1996, pp. 137175.

Alexander Gnedin, Grigori Olshanski.
"A qAnalogue of de Finetti's Theorem".
Electronic Journal of Combinatorics, 16 no. 1 (2009), R78.

Frederick M. Goodman, Sergei V. Kerov.
"The Martin Boundary of the YoungFibonacci Lattice".
Journal of Algebraic Combinatorics 11 (2000), 17–48.

Soichi Okada.
"Algebras Associated to the YoungFibonacci Lattice".
Transactions Of The
American Mathematical Society Volume 346, Number 2, 1994.

Alan D. Sokal.
"The Euler and Springer numbers as moment sequences".
Expositiones Mathematicae 38.1 (2020): 126.

S. Kerov.
"A differential model for the growth of Young diagrams".
AMS Translations (1999), 111130.

S. Kerov.
"The asymptotics of interlacing sequences and the growth of continual Young diagrams".
Journal of Mathematical Sciences 80 (1996), no. 3, 17601767.

A. Okounkov.
"Infinite wedge and random partitions".
Seleca Math. 7 (2001), no. 1, 5781.