Introduction to Asymptotic Representation Theory
What does a typical irreducible representation of the symmetric group $S_n$ look like when $n$ is very large? What is the length of the longest increasing subsequence of a random permutation? And what should it mean to take a “limit” of representations or characters as $n\to\infty$? Surprisingly, these questions are closely related. Their common language involves Young diagrams, symmetric functions, probability, and the representation theory of symmetric groups.
This course is an introduction to asymptotic representation theory, with symmetric groups as our main example. We will begin with the classical theory of Young diagrams and tableaux, representations and characters of symmetric groups, branching rules, and symmetric functions. We will then study random Young diagrams and Plancherel measure, their connection with longest increasing subsequences of random permutations through the Robinson–Schensted correspondence, and the Vershik–Kerov–Logan–Shepp limit shape.
In the final part of the course we will turn to the infinite symmetric group $S(\infty)$, positive harmonic functions on the Young graph, and the classification of extreme characters of $S(\infty)$ by the Thoma simplex.
The course will emphasize concrete examples and the ideas behind the main constructions and asymptotic results rather than maximal generality. Asymptotic representation theory lies at a crossroads of modern mathematics and mathematical physics, with connections to random matrices, integrable probability, combinatorics, and statistical mechanics. We will conclude with a glimpse of some of these connections and directions for further study.
This course is an introduction to asymptotic representation theory, with symmetric groups as our main example. We will begin with the classical theory of Young diagrams and tableaux, representations and characters of symmetric groups, branching rules, and symmetric functions. We will then study random Young diagrams and Plancherel measure, their connection with longest increasing subsequences of random permutations through the Robinson–Schensted correspondence, and the Vershik–Kerov–Logan–Shepp limit shape.
In the final part of the course we will turn to the infinite symmetric group $S(\infty)$, positive harmonic functions on the Young graph, and the classification of extreme characters of $S(\infty)$ by the Thoma simplex.
The course will emphasize concrete examples and the ideas behind the main constructions and asymptotic results rather than maximal generality. Asymptotic representation theory lies at a crossroads of modern mathematics and mathematical physics, with connections to random matrices, integrable probability, combinatorics, and statistical mechanics. We will conclude with a glimpse of some of these connections and directions for further study.
Lecturer
Date
1st September, 2026 ~ -
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday,Thursday | 13:30 - 15:05 | - | - | - |
Prerequisite
Basic linear algebra and abstract algebra, including some familiarity with finite groups. Basic probability theory at the level of a first undergraduate course will also be useful. Some familiarity with representations of finite groups is desirable but not essential. No previous knowledge of symmetric functions or Young tableaux will be assumed.
Syllabus
1. Young diagrams and tableaux. Partitions, standard and semistandard Young tableaux, the Young graph, the hook-length formula.
2. Representations of symmetric groups. Irreducible representations and characters of $S_n$, Specht modules, branching rules, examples for small symmetric groups.
3. Symmetric functions and characters. Schur functions, power sums, the Hall inner product, Frobenius characteristic, and the connection between symmetric functions and representations of $S_n$.
4. Robinson–Schensted correspondence. The insertion algorithm, longest increasing subsequences, and the relation between random permutations and Young tableaux.
5. Plancherel measure and random Young diagrams. Plancherel measure from representation theory and from random permutations, Plancherel growth, and the geometry of large random Young diagrams.
6. The limit-shape phenomenon. Scaling of Young diagrams, the asymptotic hook formula, the associated variational problem, and the Vershik–Kerov–Logan–Shepp limit-shape theorem.
7. The infinite symmetric group and coherent systems. $S(\infty)$, characters, coherent measures on the Young graph, central measures on paths, and the Pascal graph/de Finetti theorem as a model example.
8. The boundary of the Young graph. Extreme characters, the Thoma simplex and Thoma's theorem, asymptotic interpretation of the Thoma parameters.
Time permitting, we will discuss further developments and problems suggested by the material of the course.
2. Representations of symmetric groups. Irreducible representations and characters of $S_n$, Specht modules, branching rules, examples for small symmetric groups.
3. Symmetric functions and characters. Schur functions, power sums, the Hall inner product, Frobenius characteristic, and the connection between symmetric functions and representations of $S_n$.
4. Robinson–Schensted correspondence. The insertion algorithm, longest increasing subsequences, and the relation between random permutations and Young tableaux.
5. Plancherel measure and random Young diagrams. Plancherel measure from representation theory and from random permutations, Plancherel growth, and the geometry of large random Young diagrams.
6. The limit-shape phenomenon. Scaling of Young diagrams, the asymptotic hook formula, the associated variational problem, and the Vershik–Kerov–Logan–Shepp limit-shape theorem.
7. The infinite symmetric group and coherent systems. $S(\infty)$, characters, coherent measures on the Young graph, central measures on paths, and the Pascal graph/de Finetti theorem as a model example.
8. The boundary of the Young graph. Extreme characters, the Thoma simplex and Thoma's theorem, asymptotic interpretation of the Thoma parameters.
Time permitting, we will discuss further developments and problems suggested by the material of the course.
Reference
W. Fulton, Young Tableaux
D. Romik, The Surprising Mathematics of Longest Increasing Subsequences
A. Borodin and G. Olshanski, Representations of the Infinite Symmetric Group
I. G. Macdonald, Symmetric Functions and Hall Polynomials
D. Romik, The Surprising Mathematics of Longest Increasing Subsequences
A. Borodin and G. Olshanski, Representations of the Infinite Symmetric Group
I. G. Macdonald, Symmetric Functions and Hall Polynomials
Audience
Undergraduate
, Advanced Undergraduate
, Graduate
, Postdoc
Video Public
Yes
Notes Public
Yes
Language
English