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 particular, von Neumann factors of type II. We present a number of various examples. The main problem is to find the so-called {it spectral measure of the induced representation.} The complete solution of this problem is given for two-block Young subgroups and subgroups with infinitely many singletons and finitely many finite blocks of length greater than one.
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 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.
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.
Indecomposable characters of the wreath product of a torus by the infinite block-diagonal symmetric group
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine（乌克兰国家科学院B.维尔金低温物理与工程研究所）；Institute of Science and Technology Austria (ISTA)（奥地利科学与技术研究所）
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)
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.
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…
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…
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 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 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 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.
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…
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…
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.
We give a new (inductive) proof of the classical Frobenius--Young correspondence between irreducible complex representations of the symmetric group and Young diagrams, using the new approach, suggested in \cite{OV, VO}, to determining this correspondence. We also give linear relations between Kostka numbers that follow from the decomposition of the restrictions of induced representations to the previous symmetric subgroup. We consider a realization of representations induced from Young subgroups in polylinear forms and describe its relation to Specht modules.
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 consider a symmetric quiver with relations. Its (symmetric) representations of a fixed symmetric dimension vector are encoded in the (symmetric) representation varieties. The orbits by a (symmetric) base change group action are the isomorphism classes of (symmetric) representations. The symmetric orbits are induced by simply restricting the nonsymmetric orbits. However, when it comes to orbit closure relations, it is so far an open question under which assumptions they are induced. In connection with Borel orbits of 2-nilpotent matrices of classical Lie algebras, we describe an explicit exa…
nspired by the work of J$ø$rgensen [J], we define a (upper-, lower-) symmetric recollements; and give a one-one correspondence between the equivalent classes of the upper-symmetric recollements and one of the lower-symmetric recollements, of a triangulated category. Let $\m = \left(\begin{smallmatrix} A&M 0&B \end{smallmatrix}\right)$ with bimodule $_AM_B$. We construct an upper-symmetric abelian category recollement of $\m$-mod; and a symmetric triangulated category recollement of $\underline {\m\mbox{-}\mathcal Gproj}$ if $A$ and $B$ are Gorenstein and $_AM$ and $M_B$ are projective.
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 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.
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.
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 give explicit multiplicities and formulas for multiplicities of characters appearing in the decomposition of the induced character Ind^{S_{2n}}_{C_{S_{2n}}(σ)} 1_C, where σ is an n-cycle, C_{S_{2n}}(σ) is the centralizer of σ in S_{2n}, and 1_C denotes the trivial character on C_{S_{2n}}(σ).
Let $\mathbb{N}$ be a set of the natural numbers. Symmetric inverse semigroup $R_\infty$ is the semigroup of all infinite 0-1 matrices $\left[ g_{ij}\right]_{i,j\in \mathbb{N}}$ with at most one 1 in each row and each column such that $g_{ii}=1$ on the complement of a finite set. The binary operation in $R_\infty$ is the ordinary matrix multiplication. It is clear that infinite symmetric group $\mathfrak{S}_\infty$ is a subgroup of $R_\infty$. The map $\star:\left[ g_{ij}\right]\mapsto\left[ g_{ji}\right]$ is an involution on $R_\infty$. We call a function $f$ on $R_\infty$ positive definite i…
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 establish a connection between two settings of representation stability for the symmetric groups $S_n$ over $\mathbb{C}$. One is the symmetric monoidal category ${\rm Rep}(S_{\infty})$ of algebraic representations of the infinite symmetric group $S_{\infty} = \bigcup_n S_n$, related to the theory of ${\bf FI}$-modules. The other is the family of rigid symmetric monoidal Deligne categories $\underline{\rm Rep}(S_t)$, $t \in \mathbb{C}$, together with their abelian versions $\underline{\rm Rep}^{ab}(S_t)$, constructed by Comes and Ostrik. We show that for any $t \in \mathbb{C}$ the natural fu…
The induced representation ${\rm Ind}_H^GS$ of a locally compact group $G$ is the unitary representation of the group $G$ associated with unitary representation $S:H\rightarrow U(V)$ of a subgroup $H$ of the group $G$. Our aim is to develop the concept of induced representations for infinite-dimensional groups. The induced representations for infinite-dimensional groups in not unique, as in the case of a locally compact groups. It depends on two completions $\tilde H$ and $\tilde G$ of the subgroup $H$ and the group $G$, on an extension $\tilde S:\tilde H\rightarrow U(V)$ of the representation…
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…
The present paper originated from our previous study of the problem of harmonic analysis on the infinite symmetric group. This problem leads to a family {P_z} of probability measures, the z-measures, which depend on the complex parameter z. The z-measures live on the Thoma simplex, an infinite-dimensional compact space which is a kind of dual object to the infinite symmetric group. The aim of the paper is to introduce stochastic dynamics related to the z-measures. Namely, we construct a family of diffusion processes in the Toma simplex indexed by the same parameter z. Our diffusions are obtain…
After a quick review of the representation theory of the symmetric group, we give an exposition of the tools brought about by the so-called half-infinite wedge representation of the infinite symmetric group. We show how these can be applied to find the limit shapes of several distributions on partitions. We also briefly review the variational methods available to compute these limit shapes.
We give a summary of the results from Parts I-V (math.RT/9804086, math.RT/9804087, math.RT/9804088, math.RT/9810013, math.RT/9810014). Our work originated from harmonic analysis on the infinite symmetric group. The problem of spectral decomposition for certain representations of this group leads to a family of probability measures on an infinite-dimensional simplex, which is a kind of dual object for the infinite symmetric group. To understand the nature of these measures we interpret them as stochastic point processes on the punctured real line and compute their correlation functions. The cor…
We continue the study of the correlation functions for the point stochastic processes introduced in Part I (G.Olshanski, math.RT/9804086). We find an integral representation of all the correlation functions and their explicit expression in terms of multivariate hypergeometric functions. Then we define a modification (``lifting'') of the processes which results in a substantial simplification of the structure of the correlation functions. It turns out that the ``lifted'' correlation functions are given by a determinantal formula involving a kernel. The latter has the form (A(x)B(y)-B(x)A(y))/(x…
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 problem…
