Hostname: page-component-77c89778f8-m42fx Total loading time: 0 Render date: 2024-07-17T08:10:00.750Z Has data issue: false hasContentIssue false
  • English
  • Français

Certain Unitary Representations of the Infinite Symmetric Group, I

Published online by Cambridge University Press:  22 January 2016

Nobuaki Obata
Nobuaki Obata*
Affiliation:
Department of Mathematics, Faculty of Science, Nagoya University, Chikusa-ku, Nagoya 464, Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let X be the set of all natural numbers and let be the group of all finite permutations of X. The group equipped with the discrete topology, is called the infinite symmetric group. It was discussed in F. J. Murray and J. von Neumann as a concrete example of an ICC-group, which is a discrete group with infinite conjugacy classes. It is proved that the regular representation of an ICC-group is a factor representation of type II1. The infinite symmetric group is, therefore, a group not of type I. This may be the reason why its unitary representations have not been investigated satisfactorily. In fact, only few results are known. For instance, all indecomposable central positive definite functions on , which are related to factor representations of type IIl, were given by E. Thoma. Later on, A. M. Vershik and S. V. Kerov obtained the same result by a different method in and gave a realization of the representations of type II1 in. Concerning irreducible representations, A. Lieberman and G. I. Ol’shanskii obtained a characterization of a certain family of countably many irreducible representations by introducing a particular topology in However, irreducible representations have been studied not so actively as factor representations.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 105 , March 1987 , pp. 121 - 128
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1987

References

[ 1 ] Godement, R., Les fonctions de type positif et la theorie des groupes, Trans. Amer. Math. Soc, 63 (1948), 1–84.Google Scholar
[ 2 ] Lieberman, A., The structure of certain unitary representations of infinite symmetric groups, Trans. Amer. Math. Soc, 164 (1972), 189–198.Google Scholar
[ 3 ] Murray, F. J. and Neumann, J. von, On rings of operators, IV, Ann. Math., 44 (1943), 716–808.Google Scholar
[ 4 ] Ol’shanskii, G. I., New “large” groups of type I, J. Soviet Math., 18 (1982), 22–39.Google Scholar
[ 5 ] Saito, M., Représentations unitaires monomiales d’un groupe discret, en particulier du groupe modulaire, J. Math. Soc. Japan, 26 (1974), 464–482.Google Scholar
[ 6 ] Thoma, E., Die unzerlegbaren, positiv-definiten Klassenfunktionen der abzählbar unendlichen, symmetrischen Gruppe, Math. Z., 85 (1964), 40–61.Google Scholar
[ 7 ] Vershik, A. M. and Kerov, S. V., Asymptotic theory of characters of the symmetric group, Funct. Anal. Appl., 15 (1981), 246–255.CrossRefGoogle Scholar
[ 8 ] Vershik, A. M. and Kerov, S. V., Characters and factor representations of the infinite symmetric group, Soviet Math. Dokl., 23 (1981), 389–392.Google Scholar
[ 9 ] Yoshizawa, H., Some remarks on unitary representations of the free group, Osaka Math. J., 3 (1951), 55–63.Google Scholar