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A. Vershik

Publications and source records attributed to A. Vershik.

At least 55 records · Page 3Linked to original sources
arXiv2008

The Word and Geodesic Problems in Free Solvable Groups

We study the computational complexity of the Word Problem (WP) in free solvable groups $S_{r,d}$, where $r \geq 2$ is the rank and $d \geq 2$ is the solvability class of the group. It is known that the Magnus embedding of $S_{r,d}$ into matrices provides a polynomial time decision algorithm for WP in a fixed group $S_{r,d}$. Unfortunately, the degree of the polynomial grows together with $d$, so the uniform algorithm is not polynomial in $d$. In this paper we show that WP has time complexity $O(r n \log_2 n)$ in $S_{r,2}$, and $O(n^3 r d)$ in $S_{r,d}$ for $d \geq 3$. However, it turns out, that a seemingly close problem of computing the geodesic length of elements in $S_{r,2}$ is $NP$-complete. We prove also that one can compute Fox derivatives of elements from $S_{r,d}$ in time $O(n^3 r d)$, in particular one can use efficiently the Magnus embedding in computations with free solvable groups. Our approach is based on such classical tools as the Magnus embedding and Fox calculus, as well as, on a relatively new geometric ideas, in particular, we establish a direct link between Fox derivatives and geometric flows on Cayley graphs.

math.GR↗
arXiv2008

Invariant measure for the continual Cartan subgroup

We construct and study the one-parameter semigroup of $σ$-finite measures ${\cal L}^θ$, $θ>0$, on the space of Schwartz distributions that have an infinite-dimensional abelian group of linear symmetries; this group is a continual analog of the classical Cartan subgroup of diagonal positive matrices of the group $SL(n,R)$. The parameter $θ$ is the degree of homogeneity with respect to homotheties of the space, we prove uniqueness theorem for measures with given degree of homogeneity, and call the measure with degree of homogeneity equal to one the infinite-dimensional Lebesgue measure $\cal L$. The structure of these measures is very closely related to the so-called Poisson--Dirichlet measures $PD(θ)$, and to the well-known gamma process. The nontrivial properties of the Lebesgue measure are related to the superstructure of the measure PD(1), which is called the conic Poisson--Dirichlet measure -- $CPD$. This is the most interesting $σ$-finite measure on the set of positive convergent monotonic real series.

math.FA↗
arXiv2008

The Behavior of Laplace Transform of the Invariant Measure on the Hypersphere of High Dimension

We consider the sequence of the hyperspheres $M_{n,r}$ i.e. the homogeneous transitive spaces - of the Cartan subgroup $SDiag(n,\Bbb R)$ of the group $SL(n,\Bbb R), n=1 ...$, and studied the normalized limit of the corresponding sequence of the invariant measures $m_n$ on those spaces. In the case of compact groups and homogeneous spaces, as example - for classical pairs $(SO(n), S^{n-1}), n=1 ...$ - the limit of corresponding measures is the classical infinite dimensional gaussian measure - this is well-known Maxwell-Poincare lemma. Simultaneously that Gaussian measure is a unique (up to scalar) invariant measure with respect to the action of infinite orthogonal group $O(\infty)$. This coincidences means the asymptotic equivalence between grand and small canonical ensembles for the series of the pairs $(SO(n), S^{n-1})$. Our main result shows that situation for noncompact groups, for example for the case $(SDiag(n,\Bbb R),M_{n,r})$ (the definitions see below), is completely different: the limit of measures $m_n $ does not exist in literal sense, and we show that only normalized logarithmic limit of the Laplace transform of those measures does exist. In the same time there exists the measure which is invariant measure with respect to continuous analogue of Cartan subgroup of the group $GL(\infty)$ - this is so called infinite dimensional Lebesgue (see \cite{V}). This difference is an evidence of the non-equivalence between the grand and small canonical ensembles for the noncompact case.

math-ph↗
arXiv2008

Globalization of the partial isometries of metric spaces and local approximation of the group of isometries of the Urysohn space

We prove the equivalence of the two important facts about finite metric spaces and universal Urysohn metric spaces $\Bbb U$, namely theorem A and theorem B below: Theorem A (Approximation): The group of isometry $ISO(\Bbb U)$ contains everywhere dense locally finite subgroup; Theorem G(Globalization): For each finite metric space $F$ there exists another finite metric space $\bar F$ and isometric imbedding $j$ of $F$ to $\bar F$ such that isometry $j$ induces the imbedding of the group monomorphism of the group of isometries of the space $F$ to the group of isometries of space $\bar F$ and each partial isometry of $F$ can be extended up to global isometry in $\bar F$. The fact that theorem $G$, is true was announced in 2005 by author without proof, and was proved by S.Solecki in \cite{Sol} (see also \cite{P,P1}) based on the previous complicate results of other authors. The theorem is generalization of the Hrushevski's theorem about the globalization of the partial isomorphisms of finite graphs. We intend to give a constructive proof in the same spirit for metric spaces elsewhere. We also give the strengthening of homogeneity of Urysohn space and in the last paragraph we gave a short survey of the various constructions of Urysohn space including the new proof of the construction of shift invariant universal distance matrix from \cite{CV}.

math.MG↗
arXiv2007

Scaled entropy of filtrations of $σ$-fields

We study the notion of the scaled entropy of a filtration of $σ$-fields (= decreasing sequence of $σ$-fields) introduced by the first author ({V4}). We suggest a method for computing this entropy for the sequence of $σ$-fields of pasts of a Markov process determined by a random walk over the trajectories of a Bernoulli action of a commutative or nilpotent countable group (Theorems~5,~6). Since the scaled entropy is a metric invariant of the filtration, it follows that the sequences of $σ$-fields of pasts of random walks over the trajectories of Bernoulli actions of lattices (groups ${\Bbb Z}^d$) are metrically nonisomorphic for different dimensions $d$, and for the same $d$ but different values of the entropy of the Bernoulli scheme. We give a brief survey of the metric theory of filtrations, in particular, formulate the standardness criterion and describe its connections with the scaled entropy and the notion of a tower of measures.

math.PR↗
arXiv2007

Four Drafts of The Representation Theory of the Group of Infinite Matrices over Finite Fields

Preface (A.Vershik) - about these texts (3.); I.Interpolation between inductive and projective limits of finite groups with applicatons to linear groups over finite fields; II.The characters of the groups of almost triangle matrices over finite filed; III.A Law of Large Numbers for the characters of GL_n(k) over finite field k; IV.An outline of construction of factor representations of the group GLB(F_q).

math.RT↗
arXiv2007

Krein duality, positive 2-algebras, and dilation of comultiplications (To the centenary of Mark G.Krein)

The Krein--Tannaka duality for compact groups was a generalization the Pontryagin--Van Kampen duality for locally compact abelian groups and a remote predecessor of the theory of tensor categories. It is less known that it found applications in algebraic combinatorics (``Krein algebras''). Later, this duality was substantially extended: in \cite{V}, the notion of {\it involutive algebras in positive vector duality} was introduced. In this paper, we reformulate the notions of this theory using the language of bialgebras (and Hopf algebras) and introduce the class of involutive bialgebras and positive 2-algebras. The main goal of the paper is to give a precise statement of a new problem, which we consider as one of the main problems in this field, concerning the existence of dilations (embeddings) of positive 2-algebras into involutive bialgebras, or, in other words, the problem of describing subobjects in involutive bialgebras. We define two types of subobjects in the category of bialgebras, strict and nonstrict ones, and consider the corresponding embeddings (dilations) of positive 2-algebras into bialgebras. The difference between the two types of dilations is illustrated by the example of bicommutative positive 2-algebras (commutative hypergroups). The most interesting instance of our problem concerns dilations of the Hecke algebra $H_n(q)$. It seems that in this case strict dilations may exist only for $q=p^k$ (with $p$ a prime); it is not known whether nonstrict dilations exist for other $q$. We also show that the class of finite-dimensional involutive semisimple bialgebras coincides with the class of semigroup algebras of finite inverse semigroups.

math.QA↗
arXiv2006

Traces on Infinite-Dimensional Brauer Algebras

We describe the central measures for the random walk on graded graphs. Using this description, we obtain the list of all finite traces on three infinite-dimensional algebras: on the Brauer algebra, on the partition algebra, and on the walled Brauer algebra. For the first two algebras, these lists coincide with the list of all finite traces of the infinite symmetric group. For the walled Brauer algebra, the list of finite traces coincide with the list of finite traces of the square of the latter group. We introduce the operation which corresponds to the graph another graph which called "Pasclization" of the initial graph and then give the general criteria for coinsidness of the sets of traces on both graphs.

math.RT↗
arXiv2006

A new approach to the representation theory of the symmetric groups, III: Induced representations and the Frobenius--Young correspondence

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.

math.RT↗
arXiv2005

What does a generic Markov operator look like

We consider generic i.e., forming an everywhere dense massive subset classes of Markov operators in the space $L^2(X,μ)$ with a finite continuous measure. Since there is a canonical correspondence that associates with each Markov operator a multivalued measure-preserving transformation (i.e., a polymorphism), as well as a stationary Markov chain, we can also speak about generic polymorphisms and generic Markov chains. The most important and inexpected generic properties of Markov operators (or Markov chains or polymorphisms) is nonmixing and totally nondeterministicity. It was not known even existence of such Markov operators (the first example due to M.Rozenblatt). We suppose that this class coinsided with the class of special random perturbations of $K$-automorphisms. This theory is measure theoretic counterpart of the theory of nonselfadjoint contractions and its application.

math.FA↗
arXiv2005

Towards the definition of metric hyperbolicity

We introduce measure-theoretic definitions of {\it hyperbolic structure for measure-preserving automorphisms}. A wide class of $K$-automorphisms possesses a hyperbolic structure; we prove that all $K$-automorphisms have a slightly weaker structure of {\it semi-hyperbolicity}. Instead of the notions of stable and unstable foliations and other notions from smooth theory, we use the tools of the theory of polymorphisms. The central role is played by {\it polymorphisms} associated with a special invariant equivalence relation, more exactly, with a homoclinic equivalence relation. We call an automorphism with given hyperbolic structure a hyperbolic automorphism and prove that it is canonically quasi-similar to a so-called prime nonmixing polymorphism. We present a short but necessary vocabulary of polymorphisms and Markov operators from \cite{V1,V2}.

math.DS↗
arXiv2005

Gel'fand-Tsetlin algebras, expectations, inverse limits, Fourier analysis

We define a general notion of Gel'fand-Tsetlin algebras in the framework of inductive limit of the family semisimple star-algebras and the notion of genralized conditional expectations. The inmverse limit of such family with respect to generalized expectation is bimodule over inductive limit. The most important examples conserned to inductive families of group algebras of classical series of the discrete and continuous groups. I start from some remarks about distinguished role of I.M.Gel'fand's conceptions in whole area.

math.FA↗
arXiv2005

Kantorovich metric:initial history and little-known applications

The short history of L.Kantorovich's transport problem and his metric in the framework of his activity in mathmatics and economics during a long and difficult period. We did not mention the recent impetuious developement of application of the transport probelm to differential equations etc., and concentrated on three applications not known to wide audience - in measure and ergodic theory, and metric geometry and classification of mertric spaces with measures. This is a detale version of the talk of author on the Conference devoted to 90-th annyversary of Leonid.V.Kantorovich which was at St.Petersbutrg Euler International Mathematical Institute in January 2004.

math.FA↗
arXiv2005

Fluctuation of maximal particle energy of quantum ideal gas and random partitions

We investigate the limiting distribution of the fluctuations of the maximal summand in a random partition of a large integer with respect to a multiplicative statistics. We show that for a big family of Gibbs measures on partitions (so called generalized Bose--Einstein statistics) this distribution is the well-known Gumbel distribution which usually appears in the context of indepedent random variables. In particular, it means that the (properly rescaled) maximal energy of an individual particle in the grand canonical ensemble of the $d$-dimensional quantum ideal gas has the Gumbel distribution in the limit. We also apply our result to find the fluctuations of the height of a random 3D Young diagram (plane partition) and investigate the order statistics of random partitions under generalized Bose--Einstein statistics.

math-ph↗
arXiv2004

Polymorphisms, Markov processes, random perturbations of K-automorphisms

In this paper we develop the theory of {\it polymorphisms} of measure spaces, which is a generalization of the theory of measure-preserving transformations; we describe the main notions and discuss relations to the theory of Markov processes, operator theory, ergodic theory, etc. Using these tools, we solve the problem, which appeared in dynamics in the 70s: what kind of equivalence can exist between deterministic and random dynamical systems; our refining of this question is as follows: is it possible to have the quasi-similarity of a measure-preserving automorphism and a polymorphism. We prove that it is possible and the automorphism must be a $K$-automorphism and the polymorphism must be a special random perturbation of the automorphism; more exactly, the polymorphism is a random walk over stable leaves of the $K$-automorphism. The main part of our analysis concerns quasi-deterministic Markov processes and their tail (residual) $σ$-fields; such processes correspond to the most interesting class of polymorphisms: prime nonmixing polymorphisms.

math.DS↗
arXiv2004

Strange factor representations of type II_1 and pairs of dual dynamical systems

Given a pair of dynamical systems we consider a pair of commuting von Neumann factors of type 11_1. The construction is a generalization of classical von Neumann-Murrey and grouppoid construction. It gives a natural examples of factors with non-unit coupling constant.As the partial cases we obtain Connes-Riffel-Faddeev examples of the representations of rotation algebra. We give also an example for infinite symmetric group.

math.OA↗
arXiv2004

Geometrical approach to the free sovable groups

We give a topological interpretation of the free metabelian group, following the plan described in [Ver1,Ver2]. This interpretation is based on considering the Caley graph of a finitely generated group G as one-dimensional complex; its homology group with integer coefficients as G-space; and corresponding extension of the homology group determinated by the canonical cocycle. This gives a recursive approach to free solvable groups of any degree of solvability. For metabelian case we calculate second cohomology and describe all satellite groups.

math.GR↗