Random Young diagrams and tableaux

Lecure given in a Summer School in Algebraic Combinatorics in Krakow, Poland (July '22).

Abstract

Young diagrams and (standard) Young tableaux are central objects in the theories of symmetric functions and of symmetric group representations. In this lecture, we take a probabilistic viewpoint on these objects. What does a large random Youg diagram or Young tableau, taken with an appropriate probability distribution, look like? We will discuss three sets of tools which have been used to attack these questions: the entropy method, the representation theoretical approach, and (briefly) the determinantal point process approach.

Material

• My handwritten notes for most of the lectures;
• A beamer for the last lecture (brief introduction to the use of determinantal point processes in the area);
• The exercise sheet.

(Non-exhaustive) bibliography:

• Books
• D. Romik, The surprising mathematics of longest increasing subsequences. Cambridge: Cambridge University Press (2015).
• A. Hora, The limit shape problem for ensembles of Young diagrams. Tokyo: Springer (2016).
• Articles
• P. Biane. Approximate factorization and concentration for characters of sym- metric groups. Internat. Math. Res. Notices, 4:179–192, 2001.
• V. Gorin and M. Rahman. Random sorting networks: local statistics via random matrix laws. Probab. Theory Relat. Fields, 175(1-2):45–96, 2019.
• V. Ivanov and G. Olshanski. Kerov’s central limit theorem for the Plancherel mea- sure on Young diagrams. In Symmetric functions 2001: surveys of developments and perspectives, volume 74 of NATO Sci. Ser. II Math. Phys. Chem., pages 93–151. Kluwer Acad. Publ., Dordrecht, 2002
• B. F. Logan and L. A. Shepp. A variational problem for random Young tableaux. Advances in Math., 26(2):206–222, 1977.
• B. Pittel and D. Romik. Limit shapes for random square Young tableaux. Adv. in Appl. Math., 38(2):164–209, 2007.
• A. M. Vershik and S. V. Kerov. Asymptotic behavior of the Plancherel measure of the symmetric group and the limit form of Young tableaux. Dokl. Akad. Nauk SSSR, 233(6):1024–1027, 1977.