scholarly journals Certain unitary representations of the infinite symmetric group, II

1987 ◽  
Vol 106 ◽  
pp. 143-162 ◽  
Author(s):  
Nobuaki Obata
Keyword(s):  
Symmetric Group ◽  
Discrete Group ◽  
Inductive Limit ◽  
Symmetric Groups ◽  
Discrete Groups ◽  
Type I ◽  
Infinite Symmetric Group ◽  
Locally Finite ◽  
Distinctive Features ◽  
Infinite Dimensional

The infinite symmetric group is the discrete group of all finite permutations of the set X of all natural numbers. Among discrete groups, it has distinctive features from the viewpoint of representation theory and harmonic analysis. First, it is one of the most typical ICC-groups as well as free groups and known to be a group of non-type I. Secondly, it is a locally finite group, namely, the inductive limit of usual symmetric groups . Furthermore it is contained in infinite dimensional classical groups GL(ξ), O(ξ) and U(ξ) and their representation theories are related each other.

scholarly journals Certain Unitary Representations of the Infinite Symmetric Group, I

1987 ◽  
Vol 105 ◽  
pp. 121-128 ◽  
Author(s):  
Nobuaki Obata
Keyword(s):  
Symmetric Group ◽  
Regular Representation ◽  
Unitary Representations ◽  
Irreducible Representations ◽  
Type I ◽  
Discrete Topology ◽  
Type Ii ◽  
Infinite Symmetric Group ◽  
Von Neumann ◽  
Group I

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.

scholarly journals Discrete groups whose multiplier representations are type I

1981 ◽  
Vol 31 (4) ◽  
pp. 486-495 ◽  
Author(s):  
A. K. Holzherr
Keyword(s):  
Discrete Group ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Discrete Groups ◽  
Type I ◽  
Ordinary Representation ◽  
Necessary And Sufficient

AbstractWe determine necessary and sufficient conditions for the multiplier representations of a discrete group to be type I. This result extends the corresponding result for ordinary representation given by Kaniuth in [4].

scholarly journals Fluctuations of central measures on partitions

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Pierre-Loïc Mèliot
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Symmetric Group ◽  
Central Limit ◽  
Symmetric Groups ◽  
Polynomial Functions ◽  
Young Diagrams ◽  
Random Partitions ◽  
Infinite Symmetric Group ◽  
International Audience

International audience We study the fluctuations of models of random partitions $(\mathbb{P}_n,ω )_n ∈\mathbb{N}$ stemming from the representation theory of the infinite symmetric group. Using the theory of polynomial functions on Young diagrams, we establish a central limit theorem for the values of the irreducible characters $χ ^λ$ of the symmetric groups, with $λ$ taken randomly according to the laws $\mathbb{P}_n,ω$ . This implies a central limit theorem for the rows and columns of the random partitions, and these ``geometric'' fluctuations of our models can be recovered by relating central measures on partitions, generalized riffle shuffles, and Brownian motions conditioned to stay in a Weyl chamber. Nous étudions les fluctuations de modèles de partitions aléatoires $(\mathbb{P}_n,ω )_n ∈\mathbb{N}$ issus de la théorie des représentations du groupe symétrique infini. En utilisant la théorie des fonctions polynomiales sur les diagrammes de Young, nous établissons un théorème central limite pour les valeurs des caractères irréductibles $χ ^λ$ des groupes symétriques, avec $λ$ pris aléatoirement suivant les lois $\mathbb{P}_n,ω$ . Ceci implique un théorème central limite pour les lignes et les colonnes des partitions aléatoires, et ces fluctuations ``géométriques'' de nos modèles peuvent être retrouvées en reliant les mesures centrales sur les partitions, les battages généralisés de cartes, et les mouvements browniens conditionnés à rester dans une chambre de Weyl.

scholarly journals Centralizers of the infinite symmetric group

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Zajj Daugherty ◽  
Peter Herbrich
Keyword(s):  
Symmetric Group ◽  
Symmetric Functions ◽  
Symmetric Groups ◽  
Unitary Representations ◽  
Nous Obtenons ◽  
Infinite Symmetric Group ◽  
Centralizer Algebras ◽  
Partition Algebras ◽  
International Audience ◽  
Double Commutant

International audience We review and introduce several approaches to the study of centralizer algebras of the infinite symmetric group $S_{\infty}$. Our work is led by the double commutant relationship between finite symmetric groups and partition algebras; in the case of $S_{\infty}$, we obtain centralizer algebras that are contained in partition algebras. In view of the theory of symmetric functions in non-commuting variables, we consider representations of $S_{\infty}$ that are faithful and that contain invariant elements; namely, non-unitary representations on sequence spaces. Nous étudions les algèbres du centralisateur du groupe symétrique infini $S_{\infty}$, passant en revue certaines approches et en introduisant de nouvelles. Notre travail est basé sur la relation du double commutant entre le groupe symétrique fini et les algèbres de partition; dans le cas de $S_{\infty}$, nous obtenons des algèbres du centralisateur contenues dans les algèbres de partition. Compte tenu de la théorie des fonctions symétriques en variables non commutatives, nous considérons les représentations de $S_{\infty}$ qui sont fidèles et contiennent les invariants; c’est-à-dire, les représentations non unitaires sur les espaces de suites.

scholarly journals Projective limits of quantum symmetry groups and the doubling construction for Hopf algebras

2014 ◽  
Vol 17 (02) ◽  
pp. 1450012 ◽  
Author(s):  
Adam Skalski ◽  
Piotr M. Sołtan
Keyword(s):  
Hopf Algebras ◽  
Discrete Group ◽  
Inductive Limit ◽  
Symmetric Groups ◽  
Projective Limit ◽  
Symmetry Groups ◽  
C Algebras ◽  
Quantum Symmetry ◽  
Doubling Construction ◽  
Projective Limits

The quantum symmetry group of the inductive limit of C*-algebras equipped with orthogonal filtrations is shown to be the projective limit of the quantum symmetry groups of the C*-algebras appearing in the sequence. Some explicit examples of such projective limits are studied, including the case of quantum symmetry groups of the duals of finite symmetric groups, which do not fit directly into the framework of the main theorem and require further specific study. The investigations reveal a deep connection between quantum symmetry groups of discrete group duals and the doubling construction for Hopf algebras.

scholarly journals Clebsch–Gordan coefficients of discrete groups in subgroup bases

2018 ◽  
Vol 33 (10) ◽  
pp. 1850055
Author(s):  
Gaoli Chen
Keyword(s):  
Discrete Group ◽  
Irreducible Representations ◽  
Complex Conjugate ◽  
Discrete Groups

We express each Clebsch–Gordan (CG) coefficient of a discrete group as a product of a CG coefficient of its subgroup and a factor, which we call an embedding factor. With an appropriate definition, such factors are fixed up to phase ambiguities. Particularly, they are invariant under basis transformations of irreducible representations of both the group and its subgroup. We then impose on the embedding factors constraints, which relate them to their counterparts under complex conjugate and therefore restrict the phases of embedding factors. In some cases, the phase ambiguities are reduced to sign ambiguities. We describe the procedure of obtaining embedding factors and then calculate CG coefficients of the group [Formula: see text] in terms of embedding factors of its subgroups [Formula: see text] and [Formula: see text].

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