Event
Konstantin Matveev, Brandeis University
Friday, February 23, 2018 14:00to15:00
Room VCH-3830, Université Laval, CA
Macdonald polynomials, diffusivity, and Kerov conjecture.
Pioneers of the study of totally positive (all minors non-negative) matrices, Krein, Schoenberg, Karlin, were motivated by analytic questions, but the topic has later percolated to the domain of representation theory and algebraic combinatorics. This talk is about one story illustrating this interplay of algebra and analysis. In 1952 Edrei et al. used Nevanlinna theory to prove the classification of the totally positive infinite upper triangular Toeplitz matrices. In 1960s-1980s this problem was connected to the representation theory of the infinite symmetric group by Thoma, Vershik, Kerov. In 1992 Kerov has conjectured a generalization of this theorem based on the notion of Macdonald polynomials. Over the years several special cases related to different representation-theoretic settings were proved. I will talk about a recent proof of the general case, that required a new approach. One of the key difficulties was to find a replacement of the Nevanlinna theory argument. The new argument is based on certain diffusivity on the branching graph of Macdonald polynomials.