Multidimensional hypergeometric distribution, and characters of the unitary group

S. V. Kerov

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part IX, Tome 301 (2003), pp. 35-91

Résumé

The present paper is the publication of work notes by S. V. Kerov (1946–2000) written in 1993. The author introduces a multidimensional analog of the classical hypergeometric distribution. This is a probability measure $M_n$ on the set of Young diagrams contained in the rectangle with $n$ rows and $m$ columns. The fact that the expression for $M_n$ defines a probability measure is a nontrivial combinatorial identity, which is proved in various ways. Another combinatorial identity analyzed in the paper expresses a certain compatibility of the measures $M_n$ and $M_{n+1}$. A link with Selberg type integrals is also pointed out. The work is motivated by the problem of harmonic analysis on the infinite-dimensional unitary group.