Entropy of Schur–weyl Measures

Relative dimensions of isotypic components of N–th order tensor representations of the symmetric group on n letters give a Plancherel– type measure on the space of Young diagrams with n cells and at most N rows. It was conjectured by G. Olshanski that dimensions of isotypic components of tensor representations of finite symmetric groups, after appropriate normalization, converge to a constant with respect to this family of Plancherel–type measures in the limit when N √ n converges to a constant. The main result of the paper is the proof of this conjecture.