Entropy of Schur-Weyl measures

Mkrtchyan, Sevak

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014), p. 678-713 / Harvested from Numdam

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Résumé
Abstract

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.

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.

Publié le : 2014-01-01

MR 3189089 | Zbl 1290.05148

DOI : https://doi.org/10.1214/12-AIHP519
Classification: 05D40, 05E10, 20C30, 60C05

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     title = {Entropy of Schur-Weyl measures},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {50},
     year = {2014},
     pages = {678-713},
     doi = {10.1214/12-AIHP519},
     mrnumber = {3189089},
     zbl = {1290.05148},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2014__50_2_678_0}
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Mkrtchyan, Sevak. Entropy of Schur-Weyl measures. Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) pp. 678-713. doi : 10.1214/12-AIHP519. http://gdmltest.u-ga.fr/item/AIHPB_2014__50_2_678_0/

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