Plancherel measure on shifted Young diagrams. (English) Zbl 1111.05096

Ivanov, Vladimir

Kaimanovich, V. (ed.) et al., Representation theory, dynamical systems, and asymptotic combinatorics. Based on the conference, St. Petersburg, Russia, June 8–13, 2004 on the occasion of the 70th birthday of Anatoly Vershik. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-4208-0/hbk). Translations. Series 2. American Mathematical Society 217. Advances in the Mathematical Sciences 58, 73-86 (2006).

Summary: Consider random Young diagrams with a fixed number \(n\) of boxes, where the probability distribution on diagrams is determined by the Plancherel measure. As \(n\) goes to infinity, the boundary of the (suitably scaled) random diagram concentrates near a curve \(\Omega\) [B. F. Logan and L. A. Shepp, Adv. Math. 26, 206–222 (1977; Zbl 0363.62068)] and A. M. Vershik and S. Kerov [Sov. Math., Dokl. 18, 527–531 (1977); translation from Dokl. Akad. Nauk SSSR 233, 1024–1027 (1977; Zbl 0406.05008)]. In 1993, S. Kerov [C. R. Acad. Sci., Paris, Sér. I 316, No. 4, 303–308 (1993; Zbl 0793.43001)] announced a central limit theorem describing Gaussian fluctuations of random diagrams around the limit shape \(\Omega\). Here we propose an analogous theorem in the case of projective representations of symmetric groups.
For the entire collection see [Zbl 1098.22001].

MSC:

05E05	Symmetric functions and generalizations
05E10	Combinatorial aspects of representation theory
20C30	Representations of finite symmetric groups
20C32	Representations of infinite symmetric groups
60B10	Convergence of probability measures
60B15	Probability measures on groups or semigroups, Fourier transforms, factorization

Keywords:

Plancherel measure; random diagrams; projective representations of symmetric groups