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Inventiones mathematicae

, Volume 158, Issue 3, pp 551–642 | Cite as

Harmonic analysis on the infinite symmetric group

  • Sergei Kerov
  • Grigori Olshanski
  • Anatoly Vershik
Article

Abstract

The infinite symmetric group S(∞), whose elements are finite permutations of {1,2,3,...}, is a model example of a “big” group. By virtue of an old result of Murray–von Neumann, the one–sided regular representation of S(∞) in the Hilbert space ℓ2(S(∞)) generates a type II1 von Neumann factor while the two–sided regular representation is irreducible. This shows that the conventional scheme of harmonic analysis is not applicable to S(∞): for the former representation, decomposition into irreducibles is highly non–unique, and for the latter representation, there is no need of any decomposition at all. We start with constructing a compactification \(\mathfrak{S}\supset{S(\infty)}\), which we call the space of virtual permutations. Although \(\mathfrak{S}\) is no longer a group, it still admits a natural two–sided action of S(∞). Thus, \(\mathfrak{S}\) is a G–space, where G stands for the product of two copies of S(∞). On \(\mathfrak{S}\), there exists a unique G-invariant probability measure μ1, which has to be viewed as a “true” Haar measure for S(∞). More generally, we include μ1 into a family {μ t : t>0} of distinguished G-quasiinvariant probability measures on virtual permutations. By making use of these measures, we construct a family {T z : z∈ℂ} of unitary representations of G, called generalized regular representations (each representation T z with z≠=0 can be realized in the Hilbert space \(L^2(\mathfrak{S}, \mu_t)\), where t=|z|2). As |z|→∞, the generalized regular representations T z approach, in a suitable sense, the “naive” two–sided regular representation of the group G in the space ℓ2(S(∞)). In contrast with the latter representation, the generalized regular representations T z are highly reducible and have a rich structure. We prove that any T z admits a (unique) decomposition into a multiplicity free continuous integral of irreducible representations of G. For any two distinct (and not conjugate) complex numbers z1, z2, the spectral types of the representations \(T_{z_1}\) and \(T_{z_2}\) are shown to be disjoint. In the case z∈ℤ, a complete description of the spectral type is obtained. Further work on the case z∈ℂ∖ℤ reveals a remarkable link with stochastic point processes and random matrix theory.

Keywords

Symmetric Group Unitary Representation Young Diagram Spectral Type Inductive Limit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2004

Authors and Affiliations

  1. 1.Institute for Information Transmission ProblemsMoscowRussia
  2. 2.Steklov Institute of Mathematics at St. PetersburgSt. PetersburgRussia

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