Harmonic analysis on the infinite symmetric group

Abstract

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 ℓ 2 (S(∞)) generates a type II 1 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 S ⊃ S(∞), which we call the space of virtual permutations. Although S is no longer a group, it still admits a natural two-sided action of S(∞). Thus, S is a G-space, where G stands for the product of two copies of S(∞). On S, there exists a unique Ginvariant probability measure µ 1 , which has to be viewed as a "true" Haar measure for S(∞). More generally, we include µ 1 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 L 2 (S, µt), where t = |z| 2 ). As |z| → ∞, the generalized regular representations Tz approach, in a suitable sense, the "naive" two-sided regular representation of the group G in the space ℓ 2 (S(∞)). In contrast with the latter representation, the generalized regular representations Tz are highly reducible and have a rich structure. We prove that any Tz admits a (unique) decomposition into a multiplicity free continuous integral of irreducible representations of G. For any two distinct (and not conjugate) complex numbers z 1 , z 2 , the spectral types of the representations Tz 1 and Tz 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.

Asymptotics of q-plancherel measures

“…Link with the representation theory of the infinite symmetric group. This result on the first order asymptotics could alternatively have been obtained by using general results coming from the representation theory of the infinite symmetric group S ∞ , see in particular [KV81,Ker03,KOV04]. However, it seems impossible to understand the fluctuations with this approach.…”

Commutation relations and Markov chains

“…The z-measures are fundamental objects in representation theory (see [KOV1], [KOV2]) and become the Plancherel measure of the symmetric group in the limit z, z ′ → ∞.…”

A survey of amenability theory for direct-limit groups

“…See also [14]. A very natural construction has also been used by Olshanski, Borodin, Kerov, and Vershik to prove a sort of Plancherel theorem for certain direct-limit groups ( [3,13]). The basic ideas of the construction of the regular representation seem to originate from a paper by Pickrell ([20]).…”