Dualities and invariants with applications to asymptotic representation theory

In this course we will review classical invariant theory, discuss Howe duality for classical Lie groups and its extension to Lie superalgebras.

We will use the dualities for classical pairs of Lie groups to pose asymptotic questions. We will study representations of these dual pairs in the limit when ranks of the groups grow to infinity. We will then try to extend these results to Lie superalgebras. The material in this course is meant to provide deeper understanding and some missing proofs for results in my course "From free fermions to limit shapes and beyond", as well as pose new questions in asymptotic representation theory. Some connection will be made to the lectures of Pavel Nikitin "Representation theory of symmetric groups" and "Asymptotic representation theory".

We will use the dualities for classical pairs of Lie groups to pose asymptotic questions. We will study representations of these dual pairs in the limit when ranks of the groups grow to infinity. We will then try to extend these results to Lie superalgebras. The material in this course is meant to provide deeper understanding and some missing proofs for results in my course "From free fermions to limit shapes and beyond", as well as pose new questions in asymptotic representation theory. Some connection will be made to the lectures of Pavel Nikitin "Representation theory of symmetric groups" and "Asymptotic representation theory".

Lecturer

Date

12th June ~ 6th September, 2024

Schedule

Weekday | Time | Venue | Online |
---|---|---|---|

Wednesday,Friday | 15:20 - 16:55 | A3-3-201 | - |

Prerequisite

Representation theory of symmetric group, Lie groups and Lie algebras

Syllabus

1. Reminder on classical Lie algebras

2. Lie superalgebras

3. Irreducible finite-dimensional representations of simple Lie algebras

4. Schur-Weyl duality

5. Representations of Lie superalgebras

6. Classical invariant theory, first fundamental theorem for general linear group

7. First fundamental theorem for classical Lie groups

8. Howe duality for Lie superalgebras

9. Howe duality for infinite-dimensional Lie algebras

10. Insertion algorithms and algorithmic proofs of classical dualities

11. Howe duality in asymptotic representation theory, unitary representations of infinite-dimensional classical Lie groups

12. Young diagrams, tilings, lattice paths and other combinatorial realizations

13. Limit shapes

14. Asymptotic representation theory for Lie superalgebras

2. Lie superalgebras

3. Irreducible finite-dimensional representations of simple Lie algebras

4. Schur-Weyl duality

5. Representations of Lie superalgebras

6. Classical invariant theory, first fundamental theorem for general linear group

7. First fundamental theorem for classical Lie groups

8. Howe duality for Lie superalgebras

9. Howe duality for infinite-dimensional Lie algebras

10. Insertion algorithms and algorithmic proofs of classical dualities

11. Howe duality in asymptotic representation theory, unitary representations of infinite-dimensional classical Lie groups

12. Young diagrams, tilings, lattice paths and other combinatorial realizations

13. Limit shapes

14. Asymptotic representation theory for Lie superalgebras

Reference

[1] Goodman, Wallach "Symmetry, representations and invariants"

[2] Cheng, Wang "Dualities and representations of Lie superalgebras"

[3] Howe "Remarks on classical invariant theory"

[4] Howe "Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond"

[5] Olshanski "Unitary representations of infinite-dimensional pairs (G,K) and formalism of R. Howe"

[2] Cheng, Wang "Dualities and representations of Lie superalgebras"

[3] Howe "Remarks on classical invariant theory"

[4] Howe "Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond"

[5] Olshanski "Unitary representations of infinite-dimensional pairs (G,K) and formalism of R. Howe"

Audience

Graduate

Video Public

Yes

Notes Public

Yes

Language

English

Lecturer Intro

Anton Nazarov is an associate professor at Saint Petersburg State University, Russia. He completed his PhD at the department of high-energy and elementary particle physics of Saint Petersburg State University in 2012 under the supervision of Vladimir Lyakhovsky. In 2013-2014 he was a postdoc at the University of Chicago. Anton's research interests are representation theory of Lie algebras, conformal field theory, integrable systems, determinantal point processes.