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Entropy of Schur-Weyl measures. (English. French summary) Zbl 1290.05148

Summary: 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 \(\frac{N}{\sqrt{n}}\) converges to a constant. The main result of the paper is the proof of this conjecture.

MSC:

05D40 Probabilistic methods in extremal combinatorics, including polynomial methods (combinatorial Nullstellensatz, etc.)
05E10 Combinatorial aspects of representation theory
20C30 Representations of finite symmetric groups
60C05 Combinatorial probability
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