A statistical sum associated with Young diagrams. (English. Russian original) Zbl 0693.20011

Vershik, A. M.

J. Sov. Math. 47, No. 2, 2379-2386 (1989); translation from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 164, 20-29 (1987).

For \(\beta\geq 0\) let \(\Xi_ N(\beta)=\sum_{\Lambda}((N!)^{-1/2} \dim \Lambda)^{\beta}\) where the summation refers to the pairwise non- equivalent complex irreducible representations \(\Lambda\) of the symmetric group \(S_ N\) on N letters. The asymptotic behaviour of \({\mathcal S}_ N(\beta)=N^{-1/2}\log \Xi_ N(\beta)\), as \(N\to \infty\), is well-known for \(\beta =0\), \(\beta =1\) and \(\beta =2.\)
In the paper under review the author investigates the asymptotic properties of the “Gibbs measure” \(p_{\beta,N}(\Lambda)=\Xi_ N(\beta)^{-1}((N!)^{-1/2} \dim \Lambda)^{\beta}\). The limiting Gibbs measure on the space of Young diagrams is calculated for \(\beta >0\) in Theorem 1 in the same way as the case \(\beta =2\) (Plancherel measure) was analyzed by A. M. Vershik and S. V. Kerov [Dokl. Akad. Nauk SSSR 233, 1024-1027 \(=\) Sov. Math., Dokl. 18, 527-531 (1977; Zbl 0406.05008)]. Theorem 2 shows that the case \(\beta =0\) is of different character. Theorem 3 asserts that \({\mathcal S}'_ N(0)\to -\infty\), as \(N\to \infty\). Further relations are established by assuming that \(\lim_{N\to \infty} {\mathcal S}_ N(\beta)\) exists.
The appendix written A. M. Pass contains some numerical and graphic information on \({\mathcal S}_ N(\beta)\).

Reviewer: M.Szalay

MSC:

20C30	Representations of finite symmetric groups
43A65	Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
60B10	Convergence of probability measures
20P05	Probabilistic methods in group theory

Keywords:

complex irreducible representations; symmetric group; asymptotic behaviour; Gibbs measure; Young diagrams

Citations:

Zbl 0406.05008

References:

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