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Geometric and Functional Analysis

, Volume 22, Issue 4, pp 938–975 | Cite as

On the Vershik–Kerov Conjecture Concerning the Shannon–McMillan–Breiman Theorem for the Plancherel Family of Measures on the Space of Young Diagrams

  • Alexander I. Bufetov
Article

Abstract

Vershik and Kerov conjectured in 1985 that dimensions of irreducible representations of finite symmetric groups, after appropriate normalization, converge to a constant with respect to the Plancherel family of measures on the space of Young diagrams. The statement of the Vershik–Kerov conjecture can be seen as an analogue of the Shannon–McMillan–Breiman Theorem for the non-stationary Markov process of the growth of a Young diagram. The limiting constant is then interpreted as the entropy of the Plancherel measure. The main result of the paper is the proof of the Vershik–Kerov conjecture. The argument is based on the methods of Borodin, Okounkov and Olshanski.

Keywords

Bessel Function Symmetric Group Local Pattern Young Diagram Young Tableau 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.The Steklov Institute of MathematicsMoscowRussia
  2. 2.The Institute for Information Transmission ProblemsMoscowRussia
  3. 3.National Research University Higher School of EconomicsMoscowRussia
  4. 4.The Independent University of MoscowMoscowRussia
  5. 5.Rice UniversityHoustonUSA

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