06.06.2023, 15 Uhr
Phenomena in High Dimensions
Pierre Youssef (New York University, Abu Dhabi)
Pierre Cartier (Institut des Hautes Études Scientifiques, Paris)
Pierre Cartier: An introduction to the representation theory of the symmetric group
The theory of the irreducible linear representations of the symmetric group is a classical subject, with crucial contributions from FROBENIUS, around 1900, and YOUNG, around 1925. I shall give an introduction to their results, but I shall also describe an alternative construction, due to JUCY and MURPHY, around 1970. This lecture should be accessible to anyone with a good knowledge of linear algebra, as well as a modicum of mastery of (elementary) group theory. It should be considered as an introduction to the more advanced results described in the second lecture.
Pierre Cartier: Asymptotic properties of large Young diagrams: Old and new results
Young diagrams are the essential tool in the description of the representations of the symmetric group, as reviewed in the first lecture. The question asked and solved by VERSHIK forty years ago, consists in considering "random Young diagrams" and their limiting form when the number of boxes is large. We need first to explain the meaning of "random" by using elementary methods of probability theory, and to develop clever methods for the counting of some congurations (trees, maps, etc...). We shall also hint at the most recent developments of this subject, essentially by KEROV, BIANE and FERAY. For further reading, I refer to my lecture at the Bourbaki Seminar (June 2013) published in the "Asterisque" series of the French Mathematical Society.