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A. Vershik

Publications and source records attributed to A. Vershik.

At least 19 recordsLinked to original sources
arXiv2023

Classification of measurable functions of several variables and matrix distributions

We consider the notion of the matrix (tensor) distribution of a measurable function of several variables. On the one hand, it is an invariant of this function with respect to a certain group of transformations of variables; on the other hand, there is a special probability measure in the space of matrices (tensors) that is invariant under actions of natural infinite permutation groups. The intricate interplay of both interpretations of matrix (tensor) distributions makes them an important subject of modern functional analysis. We state and prove a theorem saying that, under certain conditions on a measurable function of two variables, its matrix distribution is a~complete invariant.

math.DS↗
arXiv2023

Limit spectral measures of matrix distributions of metric triples

A notion of the limit spectral measure of a metric triple (i.e., a metric measure space) is defined. If the metric is square integrable, then the limit spectral measure is deterministic and coinsides with the spectrum of the integral operator in $L^2(μ)$ with kernel $ρ$. We construct an example in which there is no deterministic spectral measure.

math.RT↗
arXiv2022

One-dimensional central measures on numberings of ordered sets

We describe one-dimensional central measures on numberings (tableaux) of ideals of partially ordered sets (posets). As the main example, we study the poset ${\Bbb Z}_+^d$ and the graph of its finite ideals, multidimensional Young tableaux; for $d=2$, it is the ordinary Young graph. The central measures are stratified by dimension; in the paper we give a complete description of the one-dimensional stratum and prove that every ergodic central measure is uniquely determined by its frequencies. The suggested method, in particular, gives the first purely combinatorial proof of E.~Thoma's theorem for one-dimensional central measures different from the Plancherel measure (which is of dimension~$2$).

math.CO↗
arXiv2022

Central Measures of Continuous Graded Graphs:\\ the Case of Distinct Frequencies

We define a class of continuous graded graphs similar to the graph of Gelfand--Tsetlin patterns, and describe the set of all ergodic central measures of discrete type on the path spaces of such graphs. The main observation is that an ergodic central measure on a subgraph of a Pascal-type graph can often be obtained as the restriction of the standard Bernoulli measure to the path space of the subgraph. This observation dramatically changes the approach to finding central measures also on discrete graphs, such as the famous Young graph. The simplest example of this type is given by the theorem on the weak limits of normalized Lebesgue measures on simplices; these are the so-called Cesàro measures, which are concentrated on the sequences with prescribed Cesàro limits (this limit parametrizes the corresponding measure). More complicated examples are the graphs of continuous Young diagrams with fixed number of rows and the graphs of spectra of infinite Hermitian matrices of finite rank. We prove existence and uniqueness theorems for ergodic central measures and describe their structure. In particular, our results 1) give a new spectral description of the so-called infinite-dimensional Wishart measures~\cite{W}~ -- ergodic unitarily invariant measures of discrete type on the set of infinite Hermitian matrices; 2) describe the structure of continuous analogs of measures on discrete graded graphs. New problems and connections which appear are to be considered in new publications.

math.CO↗
arXiv2021

The Schur--Weyl graph and Thoma's theorem

We define a graded graph, called the Schur--Weyl graph, which arises naturally when one considers simultaneously the RSK algorithm and the classical duality between representations of the symmetric and general linear groups. As one of the first applications of this graph, we give a new proof of the completeness of the list of discrete indecomposable characters of the infinite symmetric group.

math.RT↗
arXiv2019

Groups generated by involutions of diamond-shaped graphs, and deformations of Young's orthogonal form

With an arbitrary finite graph having a special form of 2-intervals (a diamond-shaped graph) we associate a subgroup of a symmetric group and a representation of this subgroup; state a series of problems on such groups and their representations; and present results of some computer simulations. The case we are most interested in is that of the Young graph and subgroups generated by natural involutions of Young tableaux. In particular, the classical Young's orthogonal form can be regarded as a deformation of our construction. We also state asymptotic problems for infinite groups.

math.RT↗
arXiv2019

The intrinsic hyperplane arrangement in an arbitrary irreducible representation of the symmetric group

For every irreducible complex representation~$π_λ$ of the symmetric group~$§_n$, we construct, in a canonical way, a so-called intrinsic hyperplane arrangement~$\A_λ$ in the space of~$π_λ$. This arrangement is a direct generalization of the classical braid arrangement (which is the special case of our construction corresponding to the natural representation of~$§_n$), has a natural description in terms of invariant subspaces of Young subgroups, and enjoys a number of remarkable properties.

math.CO↗
arXiv2019

On a universal Borel adic space

We prove that the so-called uniadic graph and its adic automorphism are Borel universal, i.e., every aperiodic Borel automorphism is isomorphic to the restriction of this automorphism to a subset invariant under the adic transformation, the isomorphism being defined on a universal (with respect to the measure) set. We develop the concept of basic filtrations and combinatorial definiteness of automorphisms suggested in our previous paper.

math.DS↗
arXiv2019

The asymptotics of the partition of the cube into Weyl simplices, and an encoding of a Bernoulli scheme

We suggest a combinatorial method of encoding continuous symbolic dynamical systems. A~continuous phase space, the infinite-dimensional cube, turns into the path space of a tree, and the shift is mapped to a transformation which was called a "transfer." The central problem is that of distinguishability: does the encoding separate almost all points of the space? The main result says that the partition of the cube into Weyl simplices satisfies this property.\footnote{{\it Keywords:} combinatorial encoding, transfer, Bernoulli scheme, graded graph.

math.CO↗
arXiv2018

Combinatorial invariants of metric filtrations and automorphisms; the universal adic graph

We suggest a combinatorial classification of metric filtrations in measure spaces; a complete invariant of such a filtration is its combinatorial scheme, a measure on the space of hierarchies of the group~$\mathbb Z$. In turn, the notion of combinatorial scheme is a source of new metric invariants of automorphisms approximated via basic filtrations. We construct a universal graph endowed with an adic structure such that every automorphism can be realized in its path space.

math.DS↗
arXiv2018

Three theorems on the uniqueness of the Plancherel measure from different viewpoints

We consider three uniqueness theorems: one from the theory of meromorphic functions, another one from asymptotic combinatorics, and the third one about representations of the infinite symmetric group. The first theorem establishes the uniqueness of the function~$\exp z$ in a class of entire functions. The second one is about the uniqueness of a random monotone nondegenerate numbering of the two-dimensional lattice~$\Bbb Z^2_+$, or of a nondegenerate central measure on the space of infinite Young tableaux. And the third theorem establishes the uniqueness of a representation of the infinite symmetric group~${\frak S}_{\infty}$ whose restrictions to finite subgroups have vanishingly few invariant vectors. But in fact all three theorems are, up to a nontrivial rephrasing of conditions from one area of mathematics in terms of another area, the same theorem! Up to now, researchers working in each of these areas were not aware of this equivalence. The parallelism of these uniqueness theorems on the one hand, and the difference of their proofs on the other hand call for a deeper analysis of the nature of uniqueness and suggest to transfer the methods of proof between the areas. More exactly, each of these theorems establishes the uniqueness of the so-called Plancherel measure, which is the main object of our paper. In particular, we show that this notion is general for all locally finite groups.

math.FA↗
arXiv2018

The absolute of finitely generated groups: II. The Laplacian and degenerate parts

The article continues the series of papers on the absolute of finitely generated groups. The absolute of a group with a fixed system of generators is defined as the set of ergodic Markov measures for which the system of cotransition probabilities is the same as for the simple (right) random walk generated by the uniform distribution on the generators. The absolute is a new boundary of a group, generated by random walks on the group. We divide the absolute into the Laplacian part and degenerate part and describe the connection between the absolute, homogeneous Markov processes, and the Laplace operator; prove that the Laplacian part is preserved under tak- ing some central extensions of groups; reduce the computation of the Laplacian part of the absolute of a nilpotent group to that of its abelinization; consider a number of fundamental examples (free groups, commutative groups, the discrete Heisenberg group). Keywords: absolute, Laplace operator, dynamic Cayley graph, nilpotent groups, Laplacian part of absolute.

math.FA↗
arXiv2018

The Absolute of finitely generated groups: I. Commutative groups

We give a complete description of the absolute of commutative finitely generated groups and semigroups. The absolute (previously called the exit boundary) is a further elaboration of the notion of the boundary of a random walk on a group (the Poisson--Furstenberg boundary); namely, the absolute of a (semi)group is the set of ergodic central measures on the compactum of all infinite trajectories of a simple random walk on the group. Related notions have been discussed in the probability literature: Martin boundary, entrance and exit boundaries (Dynkin), central measures on path spaces of graphs (Vershik--Kerov). A central measure (with respect to a finite system of generators of a group or semigroup) is a Markov measure on the space of trajectories whose cotransition distribution at every point is the uniform distribution on the generators (i.e., a measure of maximal entropy). For a more general notion of measures with a given cocycle. For the group~$\Bbb Z$, the problem of describing the absolute is solved exactly by the classical de~Finetti's theorem. The main result of this paper, which is a far-reaching generalization of de~Finetti's theorem, is as follows: the absolute of a commutative semigroup coincides with the set of central measures corresponding to (nonstationary) Markov chains with independent identically distributed increments. Topologically, the absolute is (in the main case) a closed disk of finite dimension.

math.GR↗
arXiv2017

Duality and free measures in vector spaces, the spectral theory of actions of non-locally compact groups

The paper presents a general duality theory for vector measure spaces taking its origin in the author's papers written in the 1960s. The main result establishes a direct correspondence between the geometry of a measure in a vector space and the properties of the space of measurable linear functionals on this space regarded as closed subspaces of an abstract space of measurable functions. An example of useful new features of this theory is the notion of a free measure and its applications.

math.PR↗