GRADuate International School in COmbinatorics

Here you can find the lecture notes. If you find any typos or mistakes, please let me know. If you don’t feel very comfortable with representation theory and have found this part of the course too sketchy (it definitely was very sketchy), you might want to look into slightly more detailed notes on this topic:

Here I give a quick (but more detailed) crash course on the representation theory of finite groups,

Here you can find some details on the representation theory of the symmetric group.

Here are the slides explaining the RSK algorithm. Finally,

the exercises are almost ready (warning: this file might constantly change).

• If you like this topic you might want to look at the following articles and books. My personal recommendations for books that provide a perfect introduction to this topic are denoted with ❤️):
  • P. Biane. Approximate factorization and concentration for characters of symmetric groups. Internat. Math. Res. Notices, 4 179–192, 2001.
  • M. Dołęga, V. Féray, P. Śniady Explicit combinatorial interpretation of Kerov character polynomials as numbers of permutation factorizations. Adv. Math., 225 (1), 81-120, 2010.
  • V. Ivanov and G. Olshanski. Kerov’s central limit theorem for the Plancherel measure 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.
  • ❤️ P-L. Méliot. Representation Theory of Symmetric Groups. Chapman and Hall/CRC (2017).
  • ❤️ D. Romik. The Surprising Mathematics of Longest Increasing Subsequences. Cambridge University Press, (2015).
  • ❤️ B. E. Sagan. The Symmetric Group. Springer New York, NY (2001).
  • J-P. Serre. Linear Representations of Finite Groups. Springer New York, NY (1977).
  • 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.