Entropy of Schur-Weyl measures

Mkrtchyan, Sevak

Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 2, pp. 678-713.

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.

Classification: 05D40, 05E10, 20C30, 60C05
Keywords: asymptotic representation theory, Schur-Weyl duality, Plancherel measure, Schur-Weyl measure, Vershik-Kerov conjecture

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