Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Entropy of Schur–Weyl measures

Sevak Mkrtchyan

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Abstract

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.

Résumé

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

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

Dates
First available in Project Euclid: 26 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1395856145

Digital Object Identifier
doi:10.1214/12-AIHP519

Mathematical Reviews number (MathSciNet)
MR3189089

Zentralblatt MATH identifier
1290.05148

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

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

Citation

Mkrtchyan, Sevak. Entropy of Schur–Weyl measures. Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), no. 2, 678--713. doi:10.1214/12-AIHP519. https://projecteuclid.org/euclid.aihp/1395856145


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