Asymptotic representation theory
One of the classical objects in asymptotic representation theory is the infinite symmetric group. The main objective of this course is to explore various questions and methods of the theory by examining the Plancherel measure on the set of Young diagrams, which parametrize the irreducible representations of symmetric groups. We will begin with the study of the limit shape using variational calculus techniques, followed by a discussion on the fluctuations around the limit shape. The theory of determinantal point processes provides a convenient framework for studying these fluctuations, and it also allows us to establish connections with random matrices, and also with TASEP and similar point processes.
This course serves as a continuation of the representation theory of symmetric groups course. However, we intend to review all the main concepts discussed in the previous course at the beginning of the current course. The only prerequisites are basic courses on algebra, probability and functional analysis. Additionally, we will briefly discuss the connection between this course and "From free fermions to limit shapes and beyond" by Anton Nazarov.
This course serves as a continuation of the representation theory of symmetric groups course. However, we intend to review all the main concepts discussed in the previous course at the beginning of the current course. The only prerequisites are basic courses on algebra, probability and functional analysis. Additionally, we will briefly discuss the connection between this course and "From free fermions to limit shapes and beyond" by Anton Nazarov.
Lecturer
Date
19th September ~ 19th December, 2023
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday,Thursday | 13:30 - 15:05 | A3-1a-204 | ZOOM 05 | 293 812 9202 | BIMSA |
Prerequisite
Undergaduate Algebra, Probability and Functional Analysis
Syllabus
In the course we will mainly discuss the following topics:
representations of the symmetric groups, Bratteli diagrams, Young graph, hook length formula;
patience sorting, Robinson-Schensted-Knuth algorithm, Cauchy identity, Howe duality;
Plancherel measure, hook integral, variational problem, Hilbert transform, fractional calculus, limit shape, asymptotics of the longest increasing subsequences;
fluctuations around the limit shape, determinantal point processes, sine process, Airy process, discrete Bessel kernel, poissonization and depoissonization, Tracy-Widom distribution, Baik-Deift-Johansson Theorem.
representations of the symmetric groups, Bratteli diagrams, Young graph, hook length formula;
patience sorting, Robinson-Schensted-Knuth algorithm, Cauchy identity, Howe duality;
Plancherel measure, hook integral, variational problem, Hilbert transform, fractional calculus, limit shape, asymptotics of the longest increasing subsequences;
fluctuations around the limit shape, determinantal point processes, sine process, Airy process, discrete Bessel kernel, poissonization and depoissonization, Tracy-Widom distribution, Baik-Deift-Johansson Theorem.
Reference
Vershik, A. M., and Kerov, S. V. Asymptotics of the Plancherel measure
of the symmetric group and the limiting shape of Young tableaux. Soviet Math.
Dokl., 18, 1977, pp. 527–531.
Baik, J., Deift, P., and Johansson, K. On the distribution of the length
of the longest increasing subsequence of random permutations. J. Amer. Math.
Soc., 12, 1999, pp. 1119–1178.
Romik, D. The Surprising Mathematics of Longest Increasing Subsequences.
Okounkov, A., Vershik, A. A new approach to representation theory of symmetric groups. Selecta Mathematica, New Series 2, 581, 1996.
Fulton, Young Tableau.
A.M. Vershik and S.V. Kerov, The Grothendieck group of infinite symmetric group and symmetric functions (with the elements of the theory of K0-functor for AF-algebas). In: Representations of Lie Groups and Related Topics. Advances in Contemp. Math., vol. 7 (A.M. Vershik and D.P. Zhelobenko, editors). New York, NY; London: Gordon and Breach, 1990, pp. 39–117.
of the symmetric group and the limiting shape of Young tableaux. Soviet Math.
Dokl., 18, 1977, pp. 527–531.
Baik, J., Deift, P., and Johansson, K. On the distribution of the length
of the longest increasing subsequence of random permutations. J. Amer. Math.
Soc., 12, 1999, pp. 1119–1178.
Romik, D. The Surprising Mathematics of Longest Increasing Subsequences.
Okounkov, A., Vershik, A. A new approach to representation theory of symmetric groups. Selecta Mathematica, New Series 2, 581, 1996.
Fulton, Young Tableau.
A.M. Vershik and S.V. Kerov, The Grothendieck group of infinite symmetric group and symmetric functions (with the elements of the theory of K0-functor for AF-algebas). In: Representations of Lie Groups and Related Topics. Advances in Contemp. Math., vol. 7 (A.M. Vershik and D.P. Zhelobenko, editors). New York, NY; London: Gordon and Breach, 1990, pp. 39–117.
Audience
Advanced Undergraduate
, Graduate
, Postdoc
Video Public
Yes
Notes Public
Yes
Language
English