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A. M. Vershik

Publications and source records attributed to A. M. Vershik.

At least 19 recordsLinked to original sources
arXiv2023

Dynamics of metrics in measure spaces and scaling entropy

This survey is dedicated to a new direction in the theory of dynamical systems: the dynamics of metrics in measure spaces and new (catalytic) invariants of transformations with invariant measure. A space equipped with a measure and a metric naturally consistent with each other (a metric triple, or an $mm$-space) automatically determines the notion of its entropy class, thus allowing one to construct a theory of scaling entropy for dynamical systems with invariant measure, which is different and more general compared to the Shannon-Kolmogorov theory. This possibility was hinted at by Shannon himself, but the hint went unnoticed. The classification of metric triples in terms of matrix distributions presented in this paper was proposed by M. Gromov and A. Vershik. We describe some corollaries obtained by applying this theory. A brief overview of the paper is presented in the first chapter.

math.DS↗
arXiv2017

On the decomposition of tensor representations of symmetric groups

Following the general idea of Schur--Weyl scheme and using two suitable symmetric groups (instead of one), we try to make more explicit the classical problem of decomposing tensor representations of finite and infinite symmetric groups into irreducible components.

math.RT↗
arXiv2017

Nonunitary representations of the groups of $U(p,q)$-currents for $q\geq p>1$

The purpose of this paper is to give a construction of representations of the group of currents for semisimple groups of rank greater than one. Such groups have no unitary representations in the Fock space, since the semisimple groups of this form have no nontrivial cohomology in faithful irreducible representations. Thus we first construct cohomology of the semisimple groups in nonunitary representations. The principal method is to reduce all constructions to Iwasawa subgroups (solvable subgroups of the semisimple groups), with subsequent extension to the original group. The resulting representation is realized in the so-called quasi-Poisson Hilbert space associated with natural measures on infinite-dimensional spaces. Key words: Iwasawa subgroup, cohomology, group of currents, nonunitary representations.

math.RT↗
arXiv2017

Universal adic approximation, invariant measures and scaled entropy

We define an infinite graded graph of ordered pairs and a~canonical action of the group $\mathbb{Z}$ (the adic action) and of the infinite sum of groups of order two~$\mathcal{D}=\sum_1^{\infty} \mathbb{Z}/2\mathbb{Z}$ on the path space of the graph. It is proved that these actions are universal for both groups in the following sense: every ergodic action of these groups with invariant measure and binomial generator, multiplied by a~special action (the `odometer'), is metrically isomorphic to the canonical adic action on the path space of the graph with a~central measure. We consider a~series of related problems.

math.DS↗
arXiv2015

Equivalence of the Brownian and energy representations

We consider two unitary representations of the infinite-dimensional groups of smooth paths with values in a compact Lie group. The first representation is induced by quasi-invariance of the Wiener measure, and the second representation is the energy representation. We define these representations and their basic properties, and then we prove that these representations are unitarily equivalent.

math.PR↗
arXiv2012

Special representations of nilpotent Lie groups and the associated Poisson representations of current groups

In this paper we describe the new model of the representations of the current groups with a semisimple Lie group of the rank one. In the earlier papers of 70-80-th (Araki, Gelfand-Graev-Vershik) had posed the problem about irreducible representations of the current group for $SL(2,R)$, and was used for this the well-known Fock space-structure That construction could be applied to the arbitrary locally compact group,and is based on a so called special representation of the original group $G$, with nontrivial 1-cohomology. A new construction uses the special property of one dimensional extensions (semi-direct product)of the nilpotent groups which allows immediately to produce the special representation of the group and then to apply the quasi-Poisson construction from the previous papers by authors in order to construct the representation of current group. The parabolic subgroup of the semisimple Lie group of rank one has such semidirect product, and special representation of it can be extended onto whole semisimple group. As result we obtain needed new model of the irreducible representation of semi-simple current groups.

math.RT↗
arXiv2009

Uncountable Graphs and Invariant Measures on the Set of Universal Countable Graphs

We give new examples and describe the complete lists of all measures on the set of countable homogeneous universal graphs and $K_s$-free homogeneous universal graphs (for $s\geq 3$) that are invariant with respect to the group of all permutations of the vertices. Such measures can be regarded as random graphs (respectively, random $K_s$-free graphs). The well-known example of Erdös--Rényi (ER) of the random graph corresponds to the Bernoulli measure on the set of adjacency matrices. For the case of the universal $K_s$-free graphs there were no previously known examples of the invariant measures on the space of such graphs. The main idea of our construction is based on the new notions of {\it measurable universal}, and {\it topologically universal} graphs, which are interesting themselves. The realization of the construction can be regarded as two-step randomization for universal measurable graph : {\it "randomization in vertices"} and {\it "randomization in edges"}. For $K_s$-free, $s\geq 3$ there is only randomization in vertices of the measurable graphs. The completeness of our lists is proved using the important theorem by D. Aldous about $S_{\infty}$-invariant matrices, which we reformulate in appropriate way.

math.CO↗
arXiv2008

Boundary of the braid groups and Markov--Ivanovsky normal form

We describe random walk boundaries (in particular, the Poisson--Furstenberg, or PF-boundary) for a vast family of groups in terms of the hyperbolic boundary of a special free subgroup. We prove that almost all trajectories of the random walk (with respect to an arbitrary nondegenerate measure on the group) converge to points of that boundary. This implies the stability (in the sense of \cite{Ver}) of the so-called Markov--Ivanovsky normal form for braids.

math.GT↗
arXiv2008

Integral models of unitary representations of current groups with values in semidirect products

We describe a general construction of irreducible unitary representations of the group of currents with values in the semidirect product of a locally compact subgroup $P_0$ and a one-parameter group ${\mathbb R {}}^*_+=\{r:r>0\}$ of automorphisms of $P_0$. This construction is determined by a a faithful unitary representation of $P_0$ (canonical representation) whose images under the action of the group of automorphisms tend to the identity representation as $r\to 0$. We apply this construction to the groups of currents of the maximal parabolic subgroups of the groups of motions of the $n$-dimensional real and complex Lobachevsky spaces. The obtained representations of the groups of parabolic currents can be uniquely extended to the groups of currents with values in the semisimple groups O(n,1) and U(n,1). This gives a new description of the representations of the groups of currents of these groups constructed in the 70s and realized in the Fock space. The key role in our construction is played by the so-called special representation of the parabolic subgroup $P$ and the remarkable $σ$-finite measure (Lebesgue measure) $\mathcal L$ in the space of distributions.

math.RT↗
arXiv2008

Linearly rigid metric spaces and the embedding problem

We consider the problem of isometric embedding of metric spaces to the Banach spaces; and introduce and study the remarkable class of so-called linearly rigid metric spaces: these are the spaces that admit a unique, up to isometry, linearly dense isometric embedding into a Banach space. The first nontrivial example of such a space was given by R. Holmes; he proved that the universal Urysohn space has this property. We give a criterion of linear rigidity of a metric space, which allows us to give a simple proof of the linear rigidity of the Urysohn space and some other metric spaces. The various properties of linearly rigid spaces and related spaces are considered.

math.FA↗
arXiv2008

A new approach to the representation theory of the symmetric groups. IV. $ \Bbb Z_{2}$-graded groups and algebras

We start with definitions of the general notions of the theory of $\Bbb Z_{2}$-graded algebras. Then we consider theory of inductive families of $\Bbb Z_{2}$-graded semisimple finite-dimensional algebras and its representations in the spirit of approach of the papers \cite{VO,OV} to representation theory of symmetric groups. The main example is the classical - theory of the projective representations of symmetric groups.

math.RT↗
arXiv2006

The structure of complementary series and special representations

We give a survey of several models of irreducible complementary series representations and their limits, special representations, for the groups SU(n,1) and SO(n,1), including new ones. These groups, whose geometrical meaning is well known, exhaust the list of simple Lie groups for which the identity representation is not isolated in the space of irreducible unitary representations (i.e., which do not have the Kazhdan property) and hence there exist irreducible unitary representations of these groups -- so-called ``special representations'' -- for which the first cohomology of the group with coefficients in these representations is nontrivial. By technical reasons, it is more convenient to consider the groups O(n,1) and U(n,1). Most part of the paper is devoted to the group U(n,1). The main emphasis is on the so-called commutative models of special and complementary series representations: in these models, the maximal unipotent subgroup is represented by multiplicators in the case of O(n,1), and by the canonical model of the Heisenberg representations in the case of U(n,1). Earlier, these models were studied only for the group $SL(2,\RR)$. They are especially important for realization of nonlocal representations of current groups, which will be considered elsewhere. We substantially use the ``density'' of the irreducible representations under study of SO(n,1): their restrictions to the maximal parabolic subgroup $ {P} $ are equivalent irreducible representations. Conversely, in order to extend an irreducible representation of ${P}$ to a representation of SO(n,1), we must additionally define only one involution. For the group U(n,1), the situation is similar but slightly more complicated.

math.RT↗
arXiv2006

Induced Representations of Infinite Symmetric Group

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.

math.RT↗
arXiv2005

Markov measures on Young tableaux and induced representations on the infinite symmetric group

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 studied using tensor model of those representations which alows easily to obtain the formulas for Gel'fand-Zetlin basis.

math.RT↗
arXiv2005

A New Approach to the Representation Thoery of the Symmetric Groups. 2

The new approach to the theory of complex representrations of the finite symmetric groups which based on the notions of Coxeter generators., Gelfand-Zetlin algebras, Hecke algebra, Young-Jucys-Murphi generators and which hardly used inductive structure - is systematically developed. The appearence of Young diagrams, tables is naturally explained - the set of content vectors of all Young tables is the spectra of Gel'fand-Zetlin algebra. The first steps of the theory (the list of the irrediucible representatinos, branching rule, Young orthopgonal form, Murnagan-Nakayama rule etc.)are established.This appraoch canbe used for other classical series, wreath products with symmetric groups. This paper is the revcised version of the paper of the same authors which was published in Selecta MAth. (New Series) V.2(1996).

math.RT↗
arXiv2005

The Basic Representation of the Current Group O(n,1)^X in the L^2 space over the generalized Lebesgue Measure

We give the realization of the representation of the current group O(n,1)^X where X is a manifold, in the Hilbert space of L^2(F,ν) of functionals on the the space F of the generalized functions on the manifold X which are square integrable over measure νwhich is related to a distinguish Levy process with values in R^{n-1} which generalized one dimensional gamma process. Unipotent subgroup of the group O(n,1)^X acts as the group of multiplicators. Measure νis sigma-finite and invariant under the action current group O(n-1)^X. Ther case of n=2 (SL(2,R^X)) was considered before in the series of papers starting from the article Vershik-Gel'fand-Graev (1973).

math.RT↗