A. M. Vershik

Harmonic analysis on the infinite symmetric group

AbstractThe 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

Ergodic properties of the Erdös measure, the entropy of the goldenshift, and related problems

\mathfrak{S}\supset{S(\infty)}

Some Characteristic Properties of Gaussian Stochastic Processes

, which we call the space of virtual permutations. Although

Nonholonomic problems and the theory of distributions

\mathfrak{S}

Arithmetic construction of sofic partitions of hyperbolic toral automorphisms

is no longer a group, it still admits a natural two–sided action of S(∞). Thus,

Random metric spaces and universality

\mathfrak{S}

Continuum analogues of contragredient Lie algebras (Lie algebras with a Cartan operator and nonlinear dynamical systems)

is a G–space, where G stands for the product of two copies of S(∞). On

Asymptotic combinatorics with application to mathematical physics

\mathfrak{S}

Factorial representations of path groups

, 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 {μt: t>0} of distinguished G-quasiinvariant probability measures on virtual permutations. By making use of these measures, we construct a family {Tz: z∈ℂ} of unitary representations of G, called generalized regular representations (each representation Tz with z≠=0 can be realized in the Hilbert space

Quasi-invariance of the gamma process and multiplicative properties of the Poisson-Dirichlet measures

L^2(\mathfrak{S}, \mu_t)