Point processes and the infinite symmetric group. Part I: The general formalism and the density function
Grigori Olshanski
备注：
AMSTeX, 40 pages
发表信息：
In: The orbit method in geometry and physics: in honor of A. A. Kirillov (C. Duval, L. Guieu, V. Ovsienko, eds), Progress in Math. 213. Birkhauser, 2003, pp. 349-393.
We study a 2-parametric family of probability measures on an infinite-dimensional simplex (the Thoma simplex). These measures originate in harmonic analysis on the infinite symmetric group (S.Kerov, G.Olshanski and A.Vershik, Comptes Rendus Acad. Sci. Paris I 316 (1993), 773-778). Our approach is to interpret them as probability distributions on a space of point configurations, i.e., as certain point stochastic processes, and to find the correlation functions of these processes.
In the present paper we relate the correlation functions to the solutions of certain multidimensional moment problems. Then we calculate the first correlation function which leads to a conclusion about the support of the initial measures. In the appendix, we discuss a parallel but more elementary theory related to the well-known Poisson-Dirichlet distribution.
The higher correlation functions are explicitly calculated in the subsequent paper (A.Borodin, math.RT/9804087). In the third part (A.Borodin and G.Olshanski, math.RT/9804088) we discuss some applications and relationships with the random matrix theory.
The goal of our work is to understand new phenomena in noncommutative harmonic analysis which arise when the irreducible representations depend on countably many continuous parameters.
In Part I (G.Olshanski, math.RT/9804086) and Part II (A.Borodin, math.RT/9804087) we developed an approach to certain probability distributions on the Thoma simplex. The latter has infinite dimension and is a kind of dual object for the infinite symmetric group. Our approach is based on studying the correlation functions of certain related point stochastic processes. In the present paper we consider the so-called tail point processes which describe the limit behavior of the Thoma parameters (coordinates on the Thoma simplex) with large numbers. The tail processes turn out to be stationary proc…
We study a 2-parametric family of probability measures on the space of countable point configurations on the punctured real line (the points of the random configuration are concentrated near zero). These measures (or, equivalently, point processes) have been introduced in Part II (A. Borodin, math.RT/9804087) in connection with the problem of harmonic analysis on the infinite symmetric group. The main result of the present paper is a determinantal formula for the correlation functions. The formula involves a kernel called the matrix Whittaker kernel. Each of its two diagonal blocks governs the…
We show that the class of inductive limits of the representations of finite symmetric groups with simple spectrum coinsides with the class of Markov representations of the infinite symmetric group associated with Markov measures on the space of infinite Young tableaux. We also show that the representations of infinite symmetric group induced from identity representation of two-block Young subgroup are Markov representations and find explicit formulas for transition probabilities of corresponding Markov measure on the Young diagrmas. Induced two-row representations of finite symmetric group are…
We give a complete list of indecomposable characters of the infinite symmetric semigroup. In comparison with the analogous list for the infinite symmetric group, one should introduce only one new parameter, which has a clear combinatorial meaning. The paper relies on the representation theory of the finite symmetric semigroups and the representation theory of the infinite symmetric group.
In this paper, we review the representation theory of the infinite symmetric group, and we extend the works of Kerov and Vershik by proving that the irreducible characters of the infinite symmetric group always satisfy a central limit theorem. Hence, for any point of the Thoma simplex, the corresponding measures on the levels of the Young graph have a property of gaussian concentration. By using the Robinson-Schensted-Knuth algorithm and the theory of Pitman operators, we relate these results to the properties of certain random permutations obtained by riffle shuffles, and to the behaviour of …
We classify all irreducible admissible representations of three Olshanski pairs connected to the infinite symmetric group. In particular, our methods yield two simple proofs of the classical Thoma's description of the characters of the infinite symmetric group. Also, we discuss a certain operation called mixture of representations which provides a uniform construction of all irreducible admissible representations.
The Laguerre symmetric functions introduced in the note are indexed by arbitrary partitions and depend on two continuous parameters. The top degree homogeneous component of every Laguerre symmetric function coincides with the Schur function with the same index. Thus, the Laguerre symmetric functions form a two-parameter family of inhomogeneous bases in the algebra of symmetric functions. These new symmetric functions are obtained from the N-variate symmetric polynomials of the same name by a procedure of analytic continuation. The Laguerre symmetric functions are eigenvectors of a second order…
We provide an operator algebraic proof of a classical theorem of Thoma which characterizes the extremal characters of the infinite symmetric group $\mathbb{S}_\infty$. Our methods are based on noncommutative conditional independence emerging from exchangeability and we reinterpret Thoma's theorem as a noncommutative de Finetti type result. Our approach is, in parts, inspired by Jones' subfactor theory and by Okounkov's spectral proof of Thoma's theorem, and we link them by inferring spectral properties from certain commuting squares.
We review and introduce several approaches to the study of centralizer algebras of the infinite symmetric group $S_\infty$. Our study is led by the double commutant relationships between finite symmetric groups and partition algebras; each approach produces a centralizer algebra that is contained in a partition algebra. Our goal is to incorporate invariants of $S_\infty$, which ties our work to the study of symmetric functions in non-commuting variables. We resultantly explore sequence spaces as permutation modules, which yields families of non-unitary representations of $S_\infty$.
We show that double cosets of the infinite symmetric group with respect to some special subgroups admit natural structures of semigroups. We interpret elements of such semigroups in combinatorial terms (chips, colored graphs, two-dimensional surfaces with polygonal tiling) and describe explicitly the multiplications.
We consider a product of three copies of infinite symmetric group and its representations spherical with respect to the diagonal subgroup. We show that such representations generate functors from a certain category of simplicial two-dimensional surfaces to the category of Hilbert spaces and bounded linear operators.
We study the category $\mathcal{A}$ of smooth semilinear representations of the infinite symmetric group over the field of rational functions in infinitely many variables. We establish a number of results about the structure of $\mathcal{A}$, e.g., classification of injective objects, finiteness of injective dimension, computation of the Grothendieck group, and so on. We also prove that $\mathcal{A}$ is (essentially) equivalent to a simpler linear algebraic category $\mathcal{B}$, which makes many properties of $\mathcal{A}$ transparent.
We consider the totally nonfree (TNF) action of a groups and the corresponding adjoint invariant (AD) measures on the lattices of the subgroups of the given group. The main result is the description of all adjoint-invariant and TNF-measures on the lattice of subgroups of the infinite symmetric group $S_{\Bbb N}$. The problem is closely related to the theory of characters and factor representations of groups.
We introduce and study the so-called serpentine representations of the infinite symmetric group $\sinf$, which turn out to be closely related to the basic representation of the affine Lie algebra $\widehat{\mathfrak{sl}_2}$ and representations of the Virasoro algebra. This is a new version of the manuscript of the same authors "On a relation between the basic representation of the affine Lie algebra $\widehat\sl$ and a Schur--Weyl representation of the infinite symmetric group" (arXiv:1403.1558)
The z-measures on partitions originated from the problem of harmonic analysis of linear representations of the infinite symmetric group in the works of Kerov, Olshanski and Vershik (1993, 2004). A similar family corresponding to projective representations was introduced by Borodin (1997). The latter measures live on strict partitions (i.e., partitions with distinct parts), and the z-measures are supported by all partitions. In this note we describe some combinatorial relations between these two families of measures using the well-known doubling of shifted Young diagrams.
We consider a category whose morphisms are bordisms of $n$-dimensional pseudomanifolds equipped with a certain additional structure (coloring). On the other hand, we consider the product $G$ of $(n+1)$ copies of infinite symmetric group. We show that unitary representations of $G$ produce functors from the category of $(n-1)$-dimensional bordisms to the category of Hilbert spaces and bounded linear operators.
In this note the smooth (i.e. with open stabilizers) linear and {\sl semilinear} representations of certain permutation groups (such as infinite symmetric group or automorphism group of an infinite-dimensional vector space over a finite field) are studied. Many results here are well-known to the experts, at least in the case of {\sl linear representations} of symmetric group. The presented results suggest, in particular, that an analogue of Hilbert's Theorem 90 should hold: in the case of faithful action of the group on the base field the irreducible smooth semilinear representations are one-d…
The aim of the present survey paper is to provide an accessible introduction to a new chapter of representation theory - harmonic analysis for noncommutative groups with infinite-dimensional dual space. I omitted detailed proofs but tried to explain the main ideas of the theory and its connections with other fields. The fact that irreducible representations of the groups in question depend on infinitely many parameters leads to a number of new effects which never occurred in conventional noncommutative harmonic analysis. A link with stochastic point processes is especially emphasized. The expo…
The asymptotics of the first rows and columns of random Young diagrams corresponding to extremal characters of the infinite symmetric group is studied. We consider rows and columns with linear growth in $n$, the number of boxes of random diagrams, and prove the central limit theorem for them in the case of distinct Thoma parameters. We also establish a more precise statement relating the growth of rows and columns of Young diagrams to a simple independent random sampling model. After this paper was completed, the author learned that the central limit theorem has been also proved in the work of…
We extend the definition of conical representations for Riemannian symmetric spaces to a certain class of infinite-dimensional Riemannian symmetric spaces. Using an infinite-dimensional version of Weyl's Unitary Trick, there is a correspondence between smooth representations of infinite-dimensional noncompact-type Riemannian symmetric spaces and smooth representations of infinite-dimensional compact-type symmetric spaces. We classify all smooth conical representations which are unitary on the compact-type side. Finally, a new class of non-smooth unitary conical representations appears on the c…
We construct a family of Pfaffian point processes relevant for the harmonic analysis on the infinite symmetric group. The correlation functions of these processes are representable as Pfaffians with matrix valued kernels. We give explicit formulae for the matrix valued kernels in terms of the classical Whittaker functions. The obtained formulae have the same structure as that arising in the study of symplectic ensembles of Random Matrix Theory. The paper is an extended version of the author's talk at Fall 2010 MSRI Random Matrix Theory program.
Classical Schur P-functions are the particular case of Hall-Littlewood polynomials when the parameter is equal to -1. We introduce factorial (interpolation) analogues of Schur P-functions. A dimension of a skew shifted Young diagram is the number of standard tableaux of the given shape. Also these numbers are equal up to simple factors to the decomposition coefficients of the restriction of an irreducible representation of a spin-symmetric group to a smaller spin-symmetric group. In terms of the factorial Schur P-functions we obtain an explicit formula for the dimension of a skew shifted Young…
We prove that there is a natural grading-preserving isomorphism of $\sl$-modules between the basic module of the affine Lie algebra $\widehat\sl$ (with the homogeneous grading) and a Schur--Weyl module of the infinite symmetric group $\sinf$ with a grading defined through the combinatorial notion of the major index of a Young tableau, and study the properties of this isomorphism. The results reveal new and deep interrelations between the representation theory of $\widehat\sl$ and the Virasoro algebra on the one hand, and the representation theory of $\sinf$ and the related combinatorics on the…
The paper contains a survey of train constructions for infinite symmetric groups and related groups. For certain pairs (a group $G$, a subgroup $K$), we construct categories, whose morphisms are two-dimensional surfaces tiled by polygons and colored in a certain way. A product of morphisms is a gluing of combinatorial bordisms. For a unitary representation of $G$ we assign a functor from the category of bordisms to the category of Hilbert spaces and bounded operators. The construction has numerous variations, instead of surfaces there arise also one-dimensional objects of Brauer diagram type, …
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 pa…
