arXiv subjects

Yury A. Neretin

Publications and source records attributed to Yury A. Neretin.

At least 37 records · Page 2Linked to original sources

arXiv2022

Virtual permutations and polymorhisms

There is a natural map from a symmetric group $S_n$ to a smaller symmetric group $S_{n-1}$, we write a decomposition of a permutation into a product of disjoint cycles and remove the element $n$ from this expression. For this reason there exists the inverse limit $\mathfrak{S}$ of sets $S_n$. We equip $S_n$ with the uniform distribution (or more generally with an Ewens distribution) and get a structure of a measure space on $\mathfrak{S}$ (it is called 'virtual permutations' or 'Chinese restaurant process'), a double $S_\infty\times S_\infty $ of an infinite symmetric group acts on $\mathfrak{S}$ by left and right 'multiplications'. We discuss the closure of $S_\infty\times S_\infty $ in the semigroup of polymorphisms (spreading maps with spreaded Radon--Nikodym derivatives) of $\mathfrak{S}$. We get formulas for some polymorphisms, in particular for the center of the closure. Expressions are sums of multiple convolutions of Dirichlet distributions, summation sets are certain collections of dessins d'enfant.

math.RT↗

arXiv2021

Nikolay Konstantinov and the Konstantinov System

The 'Konstantinov System' was a non-standard educational institution created by the great mathematical educator Nikolay Konstantinov (1932-2021), this 'System' worked (mainly in Moscow) in 1960-80s. We discuss some sides of technologies of the 'System': methods of teaching, organization, approaches to personal growth. In Russian.

math.HO↗

arXiv2021

Inner functions of matrix argument and conjugacy classes in unitary groups

Denote by $B_n$ the set of complex square matrices of order $n$, whose Euclidean operator norms are $<1$. Its Shilov boundary is the set $U(n)$ of all unitary matrices. A holomorphic map $B_m\to B_n$ is inner if it sends $U(m)$ to $U(n)$. On the other hand we consider a group $U(n+mj)$ and its subgroup $U(j)$ embedded to $U(n+mj)$ in a block-diagonal way ($m$ blocks $U(j)$ and a unit block of size $n$). For any conjugacy class of $U(n+mj)$ with respect to $U(j)$ we assign a 'characteristic function', which is a rational inner map $B_m\to B_n$. We show that the class of inner functions, which can be obtained as 'characteristic functions', is closed with respect to natural operations as pointwise direct sums, pointwise products, compositions, substitutions to finite-dimensional representations of general linear groups, etc. We also describe explicitly the corresponding operations on conjugacy classes.

math.CV↗

arXiv2021

On a ${\mathbb C}^2$-valued integral index transform and bilateral hypergeometric series

We discuss the spectral decomposition of the hypergeometric differential operators on the line $\mathrm{Re}\, z=1/2$. Such operators arise in the problem of decomposition of tensor products of unitary representations of the universal covering of the group $SL(2\,{\mathbb R}$. Our main purpose is a search of natural bases in generalized eigenspaces and variants of the inversion formula.

math.CA↗

arXiv2021

Some remarks on traces on the infinite-dimensional Iwahori--Hecke algebra

The infinite-dimensional Iwahori--Hecke algebras $\mathcal{H}_\infty(q)$ are direct limits of the usual finite-dimensional Iwahori--Hecke algebras. They arise in a natural way as convolution algebras of bi-invariant functions on groups $\mathrm{GLB}(\mathbb{F}_q)$ of infinite-dimensional matrices over finite-fields having only finite number of non-zero matrix elements under the diagonal. In 1988 Vershik and Kerov classified all indecomposable positive traces on $\mathcal{H}_\infty(q)$. Any such trace generates a representation of the double $\mathcal{H}_\infty(q)\otimes \mathcal{H}_\infty(q)$ and of the double $\mathrm{GLB}(\mathbb{F}_q)\times \mathrm{GLB}(\mathbb{F}_q)$. We present constructions of such representations; the traces are some distinguished matrix elements. We also obtain some (simple) general statements on relations between unitary representations of groups and representations of convolution algebras of measures bi-invariant with respect to compact subgroups.

math.RT↗

arXiv2020

On double cosets of groups $GL(n)$ with respect to subgroups of block strictly triangular matrices

We parametrize the space of double cosets of the group $GL(n,\Bbbk)$ with respect to two subgroups $T_-$, $T_+$ of block strictly triangular matrices. In Addendum, we consider the quasi-regular representation of $GL(n,\Bbb{C})$ in $L^2$ on $T_- \ GL(n,\Bbb{C})$, observe that it admits an additional group of symmetries, find the joint spectrum, and observe that it is multiplicity free.

math.GR↗

arXiv2020

Groups $GL(\infty)$ over finite fields and multiplications of double cosets

Let $\mathbb F$ be a finite field. Consider a direct sum $V$ of an infinite number of copies of $\mathbb F$, consider the dual space $V^\diamond$, i.~e., the direct product of an infinite number of copies of $\mathbb F$. Consider the direct sum ${\mathbb V}=V\oplus V^\diamond$. The object of the paper is the group $\mathbf{GL}$ of continuous linear operators in $\mathbb V$. We reduce the theory of unitary representations of $\mathbf{GL}$ to projective representations of a certain category whose morphisms are linear relations in finite-dimensional linear spaces over $\mathbb F$. In fact we consider a certain family $ Q_α$ of subgroups in $\mathbb V$ preserving two-element flags, show that there is a natural multiplication on spaces of double cosets with respect to $ Q_α$, and reduce this multiplication to products of linear relations. We show that this group has type $\mathrm{I}$ and obtain an 'upper estimate' of the set of all irreducible unitary representations of $\mathbf{GL}$.

math.RT↗

arXiv2020

Polyhomomorphisms of locally compact groups

Let $G$ and $H$ be locally compact groups with fixed two-side-invariant Haar measures. A polyhomomorphism $G\to H$ is a closed subgroup $R\subset G\times H$ with a fixed Haar measure, whose marginals on $G$ and $H$ are dominated by the Haar measures on $G$ and $H$. A polyhomomorphism can be regarded as a multi-valued map sending points to sets equipped with 'uniform' measures. For polyhomomorphsisms $G\to H$, $H\to K$ there is a well-defined product $G\to K$. The set of polyhomomorphisms $G\to H$ is a metrizable compact space with respect to the Chabauty--Bourbaki topology and the product is separately continuous. A polyhomomorphism $G\to H$ determines a canonical operator $L^2(H)\to L^2(G)$, which is a partial isometry up to scalar factor. As an example, we consider locally compact infinite-dimensional linear spaces over finite fields and examine closures of groups of linear operators in semigroups of polyendomorphisms.

math.FA↗

arXiv2020

Barnes-Ismagilov integrals and hypergeometric functions of the complex field

We examine a family ${}_pG_{q}^{\mathbb C}\big[\genfrac{}{}{0pt}{}{(a)}{(b)};z\big]$ of integrals of Mellin-Barnes type over the space ${\mathbb Z}\times {\mathbb R}$, such functions $G$ naturally arise in representation theory of the Lorentz group. We express ${}_pG_{q}^{\mathbb C}(z)$ as quadratic expressions in the generalized hypergeometric functions ${}_{p}F_{q-1}$ and discuss further properties of the functions ${}_pG_{q}^{\mathbb C}(z)$.

math.CA↗

arXiv2020

Fourier transform on the Lobachevsky plane and operational calculus

The classical Fourier transform on the line sends the operator of multiplication by $x$ to $i\frac{d}{dξ}$ and the operator of differentiation $\frac{d}{d x}$ to the multiplication by $-iξ$. For the Fourier transform on the Lobachevsky plane we establish a similar correspondence for a certain family of differential operators. It appears that differential operators on the Lobachevsky plane correspond to differential-difference operators in the Fourier-image, where shift operators act in the imaginary direction, i.e., a direction transversal to the integration contour in the Plancherel formula.

math.RT↗

arXiv2019

Kolmogorov reform of mathematical education, 1970-1980

This reform was a part of a transnational movement as 'the New Math' in the United States or the activity of the Lichnerowicz commission in France. The Russian version of these events involved the whole secondary and high school (4-10 grades). The leaders of the reformation were Alexey Markushevich and Andrey Kolmogorov. Their team consisted of best educationalists and best specialists in supplementary education. The project offered unqualified support of the Academy of Sciences and of the Ministry of education. However, the program of the reform was too optimistic, apparently some strategical ideas were unrealizable (and certainly there were no sufficient preliminary experiments for their verification). Starting September 1972 Soviet school was depressed, enormous efforts of participants of the reformation and simplification of the new school curriculum could not improve the situation. The reform was stopped by Lev Pontryagin in autumn 1980. The counterreformation required quick not-prepared steps and was not well-ordered. In 1981-1985 the heavy crisis was overcome but the mathematical education did not return to the pre-reform level.

math.HO↗

arXiv2019

On spherical unitary representations of groups of spheromorphisms of Bruhat--Tits trees

Consider an infinite homogeneous tree $T_n$ of valence $n+1$, its group $Aut(T_n)$ of automorphisms, and the group $Hie(T_n)$ of its spheromorphisms (hierarchomorphisms), i.~e., the group of homeomorphisms of the boundary of $T_n$ that locally coincide with transformations defined by automorphisms. We show that the subgroup $Aut(T_n)$ is spherical in $Hie(T_n)$, i.~e., any irreducible unitary representation of $Hie(T_n)$ contains at most one $Aut(T_n)$-fixed vector. We present a combinatorial description of the space of double cosets of $Hie(T_n)$ with respect to $Aut(T_n)$ and construct a 'new' family of spherical representations of $Hie(T_n)$. We also show that the Thompson group has $PSL(2,\mathbb{Z})$-spherical unitary

math.RT↗

arXiv2019

On the group of spheromorphisms of the homogeneous non-locally finite tree

Consider a tree $\mathbb T$, all whose vertices have countable valence; its boundary is the Baire space $\mathbb{B} \simeq\mathbb{N}^{\mathbb N}$; continued fractions expansions identify the set of irrational numbers $\mathbb{R}\setminus \mathbb{Q}$ with $\mathbb B$. Removing $k$ edges from $\mathbb T$ we get a forest consisting of copies of $\mathbb T$. A spheromorphism (or hierarchomorphism) of $\mathbb T$ is an isomorphisms of two such subforests considered as a transformation of $\mathbb T$ or of $\mathbb B$. Denote the group of all spheromorphisms by $\mathrm{Hier}({\mathbb T})$. We a show that the correspondence ${\mathbb R}\setminus{\mathbb Q}\simeq{\mathbb B}$ sends the Thompson group realized by piecewise $\mathrm{PSL}_2({\mathbb Z})$-transformations to a subgroup of $\mathrm{Hier}({\mathbb T})$. We construct some unitary representations of the group $\mathrm{Hier}({\mathbb T})$, show that the group of automorphisms $\mathrm{Aut}({\mathbb T})$ is spherical in $\mathrm{Hier}({\mathbb T})$, and describe the train (enveloping category) of $\mathrm{Hier}({\mathbb T})$.

math.RT↗

arXiv2019

An analog of the Dougall formula and of the de Branges--Wilson integral

We derive a beta-integral over $\mathbb{Z}\times \mathbb{R}$ , which is a counterpart of the Dougall $_5H_5$-formula and of the de Branges--Wilson integral, our integral includes $_{10}H_{10}$-summation. For a derivation we use a two-dimensional integral transform related to representations of the Lorentz group, this transform is a counterpart of the Olevskii index transform (a synonym: Jacobi transform).

math.CA↗

arXiv2019

Description of unitary representations of the group of infinite $p$-adic integer matrices

We classify irreducible unitary representations of the group of all infinite matrices over a $p$-adic field ($p\ne 2$) with integer elements equipped with a natural topology. Any irreducible representation passes through a group $GL$ of infinite matrices over a residue ring modulo $p^k$. Irreducible representations of the latter group are induced from finite-dimensional representations of certain open subgroups.

math.RT↗

arXiv2019

A pair of commuting hypergeometric operators on the complex plane and bispectrality

We consider the standard hypergeometric differential operator $D$ regarded as an operator on the complex plane $C$ and the complex conjugate operator $\overline D$. These operators formally commute and are formally adjoint one to another with respect to an appropriate weight. We find conditions when they commute in the Nelson sense and write explicitly their joint spectral decomposition. It is determined by a two-dimensional counterpart of the Jacobi transform (synonyms: generalized Mehler--Fock transform, Olevskii transform). We also show that the inverse transform is an operator of spectral decomposition for a pair of commuting difference operators defined in terms of shifts in imaginary direction.

math.FA↗

arXiv2019

On the group of infinite $p$-adic matrices with integer elements

Let $G$ be an infinite-dimensional real classical group containing the complete unitary group (or complete orthogonal group) as a subgroup. Then $G$ generates a category of double cosets (train) and any unitary representation of $G$ can be canonically extended to the train. We prove a technical lemma about the complete group $GL$ of infinite $p$-adic matrices with integer coefficients, this lemma implies that the phenomenon of automatic extension of unitary representations to trains is valid for infinite-dimensional $p$-adic groups.

math.RT↗

arXiv2018

Restriction of representations of $GL(n+1,C)$ to $GL(n,C)$ and action of the Lie overalgebra

Consider a restriction of an irreducible finite dimensional holomorphic representation of $\GL(n+1,C)$ to the subgroup $GL(n,C)$ (it is determined by the Gelfand-Tsetlin branching rule). We write explicitly formulas for generators of the Lie algebra $gl(n+1)$ in the direct sum of representations of $\GL(n,C)$. Nontrivial generators act as differential-difference operators, the differential part has order $(n-1)$, the difference part acts on the space of parameters (highest weights) of representations. We also formulate a conjecture about unitary principal series of $GL(n,C)$

math.RT↗