Harmonic analysis on the infinite symmetric group

By S. Kerov, Grigori Olshanski, Anatoly M. Vershik. 2003

163 citations recorded by the source corpus.

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 $\mathfrak{S}\supset{S(\infty)}$, 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 {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 $L^2(\mathfrak{S}, \mu_t)$, 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 ℓ2(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 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.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/e025f2cee6074e2dbc7283570b4380b70f236df0

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Citations in the corpus

Representations of the Infinite Symmetric Group (2016, methodology): “These two cases are studied separately and using different means in Kerov, Olshanski, and Vershik [67], and in Borodin and Olshanski [9], respectively.”
How instanton combinatorics solves Painlevé VI, V and IIIs (2013, unknown): “The Whittaker kernel [9, 10] emerges in the limit KW(x, y) = lim η′→∞ 1 η′ KG ( 1 − x η′ , 1 − y η′ ) .”
Combinatorial Stochastic Processes (2006, background): “The Chinese Restaurant Process and associated computations with random partitions have found applications in Bayesian statistics [109, 287, 210, 212], and in the theory of representations of the infinite symmetric group [243].”
Vertex Operators and the Class Algebras of Symmetric Groups (2001, background): “Therefore, L1 corresponds to the map considered in [12], and L2 amounts to erasing n and n − 1 if they are both in the same cycle, or both fixed points.”
A New Approach to the Representation Theory of the Symmetric Groups. II (2005, methodology): “For the representation theory of the infinite symmetric group, see also [20, 32, 3, 21 ].”
Infinite-dimensional diffusions as limits of random walks on partitions (2007, background): “The former case was studied in [ KOV2 ]; it turns out that for each z ∈ Z, Pz is supported by a countable union of finite–dimensional faces of the simplex . The latter case, z ∈ C\Z, which is of primary interest for us, is more complex.”