Results 1 
6 of
6
Induced Representations of the Infinite Symmetric Group
"... Abstract: We study the representations of the infinite symmetric group induced from the identity representations of Young subgroups. It turns out that such induced representations can be either of type I or of type II. Each Young subgroup corresponds to a partition of the set of positive integers; d ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
Abstract: We study the representations of the infinite symmetric group induced from the identity representations of Young subgroups. It turns out that such induced representations can be either of type I or of type II. Each Young subgroup corresponds to a partition of the set of positive integers; depending on the sizes of blocks of this partition, we divide Young subgroups into two classes: large and small subgroups. The first class gives representations of type I, in particular, irreducible representations. The most part of Young subgroups of the second class give representations of type II and, in particular, von Neumann factors of type II. We present a number of various examples. The main problem is to find the socalled spectral measure of the induced representation. The complete solution of this problem is given for twoblock Young subgroups and subgroups with infinitely many singletons and finitely many finite blocks of length greater than one. Contents
Constructing Irreducible Representations of Discrete Groups
, 1996
"... . The decomposition of unitary representations of a discrete group obtained by induction from a subgroup involves commensurators. In particular Mackey has shown that quasiregular representations are irreducible if and only if the corresponding subgroups are selfcommensurizing. The purpose of this ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
. The decomposition of unitary representations of a discrete group obtained by induction from a subgroup involves commensurators. In particular Mackey has shown that quasiregular representations are irreducible if and only if the corresponding subgroups are selfcommensurizing. The purpose of this work is to describe general constructions of pairs of groups \Gamma 0 ! \Gamma with \Gamma 0 its own commensurator in \Gamma: These constructions are then applied to groups of isometries of hyperbolic spaces and to lattices in algebraic groups. 1. Introduction Let G be a separable locally compact group. The unitary dual G of G is the set of equivalence classes of irreducible representations of G; together with its Mackey Borel structure. In this paper, "representation" means "continuous unitary representation in a separable Hilbert space". Let us recall the definition of this structure [Dix, 18.5]. For each n 2 f1; 2; : : : ; 1g; let Irr n (G) denote the space of all irreducible representa...
Induced representations of . . .
, 2006
"... We study the representations of the infinite symmetric group induced from the identity representations of Young subgroups. It turns out that such induced representations can be either of type I or of type II. Each Young subgroup corresponds to a partition of the set of positive integers; depending o ..."
Abstract
 Add to MetaCart
We study the representations of the infinite symmetric group induced from the identity representations of Young subgroups. It turns out that such induced representations can be either of type I or of type II. Each Young subgroup corresponds to a partition of the set of positive integers; depending on the sizes of blocks of this partition, we divide Young subgroups into two classes: large and small subgroups. The first class gives representations of type I, in particular, irreducible representations. The most part of Young subgroups of the second class give representations of type II and, in particular, von Neumann factors of type II. We present a number of various examples. The main problem is to find the socalled spectral measure of the induced representation. The complete solution of this problem is given for twoblock Young subgroups and subgroups with infinitely many singletons and finitely many finite blocks of length greater than one.
GENERALISED HECKE ALGEBRAS AND C∗COMPLETIONS
, 2007
"... For a Hecke pair (G, H) and a finitedimensional representation σ of H on Vσ with finite range we consider a generalised Hecke algebra Hσ(G, H), which we study by embedding the given Hecke pair in a Schlichting completion (Gσ, Hσ) that comes equipped with a continuous extension σ of Hσ. There is a ..."
Abstract
 Add to MetaCart
For a Hecke pair (G, H) and a finitedimensional representation σ of H on Vσ with finite range we consider a generalised Hecke algebra Hσ(G, H), which we study by embedding the given Hecke pair in a Schlichting completion (Gσ, Hσ) that comes equipped with a continuous extension σ of Hσ. There is a (nonfull) projection pσ ∈ Cc(Gσ, B(Vσ)) such that Hσ(G, H) is isomorphic to pσCc(Gσ, B(Vσ))pσ. We study the structure and properties of C ∗completions of the generalised Hecke algebra arising from this corner realisation, and via MoritaFellRieffel equivalence we identify, in some cases explicitly, the resulting proper ideals of C ∗ (Gσ, B(Vσ)). By letting σ vary, we can compare these ideals. The main focus is on the case with dim σ = 1 and applications include ax + bgroups and the Heisenberg group.