# MATH 8852: Asymptotic Representation Theory

E-mail 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 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).

Syllabus items

# Assessment

There are two components in the course grade:
• (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 (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:
• 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: Schur-Weyl 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. Finite-dimensional 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 Vershik-Kerov'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 finite-dimensional 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. q-analogues of Pascal and Young graphs
• 9.1 Branching graphs with edge multiplicities
• Lecture 16, 10/20
• 9.2 Exchangeability and q-exchangeability
• 9.3 q-Pascal graph and its boundary
• Lecture 17, 10/25
• 9.4 q-Pascal graph and infinite Grassmannian over a finite field
10. Young-Fibonacci graph
• 10.1 Differential posets and the commutation relation
• 10.2 Young-Fibonacci 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 Young-Fibonacci 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:
• 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

Books

Papers