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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. 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