  Vershik and Kerov gave asymptotical bounds for the maximal and the typical
dimensions of irreducible representations of symmetric groups $S_n$. It was
conjectured by G. Olshanski that the maximal and the typical dimensions of the
isotypic components of tensor representations of the symmetric group admit
similar asymptotical bounds. The main result of this article is the proof of
this conjecture. Consider the natural representation of $S_n$ on
$(\mathbb{C}^N)^{\otimes n}$. Its isotypic components are parametrized by Young
diagrams with $n$ cells and at most $N$ rows. P. Biane found the limit shape of
Young diagrams when $n\rightarrow\infty,\ \sqrt{n}/N\rightarrow c$. By showing
that this limit shape is the unique solution to a variational problem, it is
proven here, that after scaling, the maximal and the typical dimensions of
isotypic components lie between positive constants. A new proof of Biane's
limit-shape theorem is obtained.
