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Relative dimensions of the isotypic components of the N-th order tensor representations of the symmetric group on n letters define a Plancherel-type measure, called the Schur-Weyl measure, on the space of Young diagrams with n cells and at most N rows. We obtain logarithmic, order-sharp bounds for the maximal dimensions of the isotypic components of the tensor representations, and prove that the typical dimensions, after appropriate normalization, converge to a constant with respect to the Schur-Weyl measures.