Representation theory of the infinite symmetric group
This course explores the representation theory of infinite symmetric groups, a fascinating extension of the well-known representation theory of finite groups. Representation theory traditionally relies on character theory, a method pioneered by G. Frobenius in the early 20th century and later expanded by H. Weyl for classical Lie groups. This course demonstrates that a similar technique can be developed for infinite-dimensional analogs of these groups, particularly focusing on the infinite symmetric group—the group of all finite permutations of the natural numbers.
The course begins with a comprehensive introduction to Thoma’s Theorem, a central result that classifies the extreme characters (analogs of the irreducible characters) of the infinite symmetric group. This theorem will be explored from various perspectives, revealing connections with other branches of mathematics, including:
- The classification of totally positive sequences,
- The classification of Schur-positive multiplicative functionals on the algebra of symmetric functions,
- The description of the entrance boundary for certain Markov chains related to the Young graph.
While the first part of the course primarily focuses on character theory, as it provides sufficient tools for addressing classification problems (similar to the case of finite groups), the second part delves into the study of unitary representations of the infinite symmetric group. Time permitting, the course will conclude with a brief discussion on the connections between the developed theory and probabilistic models in mathematical physics of representation theoretical origin.
Prerequisites: Participants should have a basic understanding of functional analysis and probability theory. While prior knowledge of the representation theory of finite groups is beneficial, it is not mandatory. The course will also utilize concepts from the representation theory of finite symmetric groups and the theory of symmetric functions; however, all necessary background will be provided. Previous coursework in these areas, such as courses offered in the previous academic year, may ease comprehension but is not required.
The course begins with a comprehensive introduction to Thoma’s Theorem, a central result that classifies the extreme characters (analogs of the irreducible characters) of the infinite symmetric group. This theorem will be explored from various perspectives, revealing connections with other branches of mathematics, including:
- The classification of totally positive sequences,
- The classification of Schur-positive multiplicative functionals on the algebra of symmetric functions,
- The description of the entrance boundary for certain Markov chains related to the Young graph.
While the first part of the course primarily focuses on character theory, as it provides sufficient tools for addressing classification problems (similar to the case of finite groups), the second part delves into the study of unitary representations of the infinite symmetric group. Time permitting, the course will conclude with a brief discussion on the connections between the developed theory and probabilistic models in mathematical physics of representation theoretical origin.
Prerequisites: Participants should have a basic understanding of functional analysis and probability theory. While prior knowledge of the representation theory of finite groups is beneficial, it is not mandatory. The course will also utilize concepts from the representation theory of finite symmetric groups and the theory of symmetric functions; however, all necessary background will be provided. Previous coursework in these areas, such as courses offered in the previous academic year, may ease comprehension but is not required.
Lecturer
Date
12th September ~ 12th December, 2024
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday,Thursday | 13:30 - 15:05 | A3-2-303 | ZOOM 05 | 293 812 9202 | BIMSA |
Prerequisite
Undergaduate Algebra, Probability and Functional Analysis
Reference
Borodin and G. Olshanski, Z-Measures on partitions, Robinson–Schensted–Knuth correspondence, and β = 2 ensembles. In: Random Matrix Models and their Applications (P.M. Bleher and A.R. Its, eds). MSRI Publications, vol. 40, Cambridge University Press, 2001, pp. 71–94.
A. Borodin, G. Olshansky. Representations of the infinite symmetric group. Cambridge University Press, 2017.
I.G. Macdonald, Symmetric Functions and Hall Polynomials. 2nd edition. Oxford University Press, 1995.
A. Okounkov, Thoma’s theorem and representations of the infinite bisymmetric group. Funct. Anal. Appl. 28 (1994), no. 2, 100–107.
G. Olshanski, An introduction to harmonic analysis on the infinite symmetric group. In: Asymptotic Combinatorics with Applications to Mathematical Physics (A. Vershik, ed). Springer Lecture Notes in Math. 1815, 2003, pp. 127–160.
A.M. Vershik and S.V. Kerov, Asymptotic theory of characters of the symmetric group. Funct. Anal. Appl. 15 (1981), no. 4, 246–255.
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.
A. Borodin, G. Olshansky. Representations of the infinite symmetric group. Cambridge University Press, 2017.
I.G. Macdonald, Symmetric Functions and Hall Polynomials. 2nd edition. Oxford University Press, 1995.
A. Okounkov, Thoma’s theorem and representations of the infinite bisymmetric group. Funct. Anal. Appl. 28 (1994), no. 2, 100–107.
G. Olshanski, An introduction to harmonic analysis on the infinite symmetric group. In: Asymptotic Combinatorics with Applications to Mathematical Physics (A. Vershik, ed). Springer Lecture Notes in Math. 1815, 2003, pp. 127–160.
A.M. Vershik and S.V. Kerov, Asymptotic theory of characters of the symmetric group. Funct. Anal. Appl. 15 (1981), no. 4, 246–255.
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