## Journal of the Mathematical Society of Japan

### Limits of characters of wreath products of a compact group $\bm{T}$ with the symmetric groups and characters of $\bm{\mathfrak{S}_\infty(T)}$ , II -- From a viewpoint of probability theory

#### Abstract

This paper is the second part of our study on limiting behavior of characters of wreath products $\mathfrak{S}_n(T)$ of compact group $T$ as $n\to\infty$ and its connection with characters of $\mathfrak{S}_\infty(T)$ . Contrasted with the first part, which has a representation-theoretical flavor, the approach of this paper is based on probabilistic (or ergodic-theoretical) methods. We apply boundary theory for a fairly general branching graph of infinite valencies to wreath products of an arbitrary compact group $T$ . We show that any character of $\mathfrak{S}_\infty(T)$ is captured as a limit of normalized irreducible characters of $\mathfrak{S}_n(T)$ as $n\to\infty$ along a path on the branching graph of $\mathfrak{S}_\infty(T)$ . This yields reconstruction of an explicit character formula for $\mathfrak{S}_\infty(T)$ .

#### Article information

Source
J. Math. Soc. Japan Volume 60, Number 4 (2008), 1187-1217.

Dates
First available in Project Euclid: 5 November 2008

http://projecteuclid.org/euclid.jmsj/1225894038

Digital Object Identifier
doi:10.2969/jmsj/06041187

Mathematical Reviews number (MathSciNet)
MR2467875

Zentralblatt MATH identifier
05500756

#### Citation

HORA, Akihito; HIRAI, Takeshi; HIRAI, Etsuko. Limits of characters of wreath products S n ( T ) of a compact group T with the symmetric groups and characters of S ∞ ( T ) , II -- From a viewpoint of probability theory. J. Math. Soc. Japan 60 (2008), no. 4, 1187--1217. doi:10.2969/jmsj/06041187. http://projecteuclid.org/euclid.jmsj/1225894038.

#### References

• A. Borodin and G. Olshanski, Harmonic functions on multiplicative graphs and interpolation polynomials, Electronic J. Combinatorics, 7 (2000), $\sharp$R28.
• R. Boyer, Character theory of infinite wreath products, Int. J. Math. Math. Sci., 2005 (2005), 1365–1379.
• R. Durrett, Probability: Theory and Examples, Duxbury Press, Belmont, California, 1991.
• E. Hewitt and K. A. Ross, Abstract Harmonic Analysis II, Springer-Verlag, Berlin Heidelberg, 1970.
• T. Hirai and E. Hirai, Character formula for wreath products of compact groups with the infinite symmetric group, Proceedings of 25th QP Conference, Quantum Probability and Related Topics in Bedlewo, Banach Center Publications, 73, Institute of Mathematics, Polish Academy of Sciences, 2006, pp.,207–221.
• T. Hirai and E. Hirai, Characters of wreath products of compact groups with the infinite symmetric group and characters of their canonical subgroups, J. Math. Kyoto Univ., 47 (2007), 269–320.
• T. Hirai, E. Hirai and A. Hora, Realizations of factor representations of finite type with emphasis on their characters for wreath products of compact groups with the infinite symmetric group, J. Math. Kyoto Univ., 46 (2006), 75–106.
• T. Hirai, E. Hirai and A. Hora, Limits of characters of wreath products $\mathfrak{S}_n(T)$ of a compact group $T$ with the symmetric groups and characters of $\mathfrak{S}_\infty(T)$, I, to appear in Nagoya Math. J.
• A. Hora and N. Obata, Quantum Probability and Spectral Analysis of Graphs, Theoretical and Mathematical Physics, Springer, 2007.
• V. Ivanov and G. Olshanski, Kerov's central limit theorem for the Plancherel measure on Young diagrams, (ed. S. Fomin), Symmetric functions 2001, Kluwer Academic Publishers, 2002, pp.,93–151.
• G. James and A. Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and Its Applications, 16, Addison-Wesley Publishing Company, Massachusetts, 1981.
• S. Kerov, The boundary of Young lattice and random Young tableaux, DIMACS Series in Discrete Math. Theoret., Computer Science, 24 (1996), 133–158.
• S. Kerov, Asymptotic representation theory of the symmetric group and its applications in analysis, Amer. Math. Soc., Providence, RI, 2003.
• S. Kerov, A. Okounkov and G. Olshanski, The boundary of the Young graph with Jack edge multiplicities, Internat. Math. Res. Notices, 1998 (1998), 173–199.
• F. D. Murnaghan, The theory of group representations, Dover Publications, Mineola, N.Y., 1963.
• P. Śniady, Gaussian fluctuations of representations of wreath products, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 9 (2006), 529–546.
• A. Vershik and S. Kerov, Asymptotic theory of characters of the symmetric group, Funct. Anal. Appl., 15 (1981), 246–255.
• Y. Yamasaki, Measures on infinite-dimensional spaces, vol.,1 and vol.,2, Kinokuniya-shoten, Tokyo, 1978.