Abstract

This course is dedicated to the thorough introduction of the subjects, mentioned in my talk at the BIMSA Integrable Systems Seminar https://researchseminars.org/talk/BIMSA-ISS/1/

In this course we will start from the infinite-wedge formalism and boson-fermion correspondence, as described in the Chapter 14 of Victor Kac's book "Infinite dimensional Lie algebras". We will use this formalism to derive the limit shape for Young diagrams for the Plancherel measure, as was done in the works of Andrei Okounkov. Then we will consider the measures on the diagrams related to the skew Howe duality. After studying GL(n)-GL(k) skew Howe duality we will consider other classical series of Lie groups. Connections to the determinantal point processes, orthogonal polynomials, Riemann-Hilbert problem and integrable systems will be discussed.

Prerequisite

The knowledge of linear algebra and basic methods of analysis (integral calculus, theory of the function of a complex variable) is required.

Some knowledge of Quantum Mechanics and Statistical Mechanics would be useful. The knowledge of representation theory of the symmetric group and Lie groups would be useful.

Syllabus

1. Infinite wedge representation

2. Free fermions and bosonization

3. Vertex operators

4. Partitions and Schur polynomials

5. Boson-fermion correspondence

6. Plancherel measure on partitions

7. Correlation kernel from free fermions and limit shape for Plancherel measure

8. Relation of Plancherel measure to RSK algorithm

9. Representations of GL(n), exterior powers and exterior algebra

10. Skew Howe GL(n)-GL(k) duality

11. Probability measure on Young diagrams in the box and dual RSK algorithm

12. Limit shape for skew GL(n)-GL(k) duality from free fermions

13. Skew Howe dualities for other classical series of Lie groups

14. Young diagrams and tableaux for symplectic groups

15. Proctor algorithm for symplectic groups

16. Limit shapes for symplectic groups and scalar Riemann-Hilbert problem

17. Local asymptotics of correlation kernel for skew GL(n)-GL(k) duality, Airy process, local fluctuations around limit shape

18. Global fluctuations around limit shape, Krawtchouk and q-Krawtchouk orthogonal polynomials, Central limit theorem

19. Asymptotics of orthogonal polynomials, Riemann-Hilbert problem and its integrability

Reference

1. Kac V.G. - Infinite dimensional Lie algebras-CUP (1995)

2. Okounkov, Andrei. "Infinite wedge and random partitions." Selecta Mathematica 7.1 (2001): 57.

3. Okounkov, Andrei. "Symmetric functions and random partitions." Symmetric functions 2001: surveys of developments and perspectives. Springer Netherlands, 2002.

4. Borodin, Alexei, and Vadim Gorin. "Lectures on integrable probability." Probability and statistical physics in St. Petersburg 91 (2016): 155-214.

5. Proctor, Robert A. "Reflection and algorithm proofs of some more Lie group dual pair identities." Journal of Combinatorial Theory, Series A 62.1 (1993): 107-127.

6. Nazarov, A., P. Nikitin, and D. Sarafannikov. "Skew Howe duality and q-Krawtchouk polynomial ensemble." Representation theory, dynamical systems, combinatorial methods. Part XXXIV, Zap. Nauchn. Sem. POMI 517: 106-124.

7. Betea, Dan, Anton Nazarov, and Travis Scrimshaw. "Limit shapes for skew Howe duality." arXiv preprint arXiv:2211.13728 (2022).

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.