Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Entropy of Schur–Weyl measures

Sevak Mkrtchyan

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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.


Les dimensions relatives des composants isotypiques des représentations tensorielles du $N$ième ordre du groupe symétrique sur $n$ lettres induisent une mesure du type Plancherel sur l’espace des diagrammes de Young avec $n$ cellules et au plus $N$ rangs. G. Olshanski a conjecturé que ces dimensions, après renormalisation, convergent vers une constante sous cette famille de mesures du type Plancherel dans la limite où $\frac{N}{\sqrt{n}}$ converge vers une constante. Le principal résultat de cet article est la preuve de cette conjecture.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 50, Number 2 (2014), 678-713.

First available in Project Euclid: 26 March 2014

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Primary: 05D40: Probabilistic methods 05E10: Combinatorial aspects of representation theory [See also 20C30] 20C30: Representations of finite symmetric groups 60C05: Combinatorial probability

Asymptotic representation theory Schur–Weyl duality Plancherel measure Schur–Weyl measure Vershik–Kerov conjecture


Mkrtchyan, Sevak. Entropy of Schur–Weyl measures. Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), no. 2, 678--713. doi:10.1214/12-AIHP519.

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