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Oleg Ogievetsky

Publications and source records attributed to Oleg Ogievetsky.

At least 19 recordsLinked to original sources
arXiv2025

Cayley--Hamilton Theorem for Orthogonal Quantum Matrix Algebras

For a family of the orthogonal $O(k)$ type Quantum Matrix algebras we establish an analogue of the Cayley--Hamilton theorem. The form of the Cayley-Hamilton identity is different in three cases. First, the cases of odd ($k=2\ell -1$) and even ($k=2\ell$) heights are different. Second, for even height orthogonal Quantum Matrix algebra we derive two versions of the Cayley--Hamilton theorem, one for its positive component $O^+(2\ell)$ and another one for the negative component $O^-(2\ell)$. In each case we introduce the spectral parameterization of the coefficients of the Cayley--Hamilton identity by the `eigenvalues' of the quantum matrices.

math.QA↗
arXiv2025

Reciprocal relations for orthogonal quantum matrices

For the family of the orthogonal quantum matrix algebras we investigate the structure of their characteristic subalgebras -- special commutative subalgebras, which for the subfamily of the reflection equation algebras appear to be central. In [OP1] we described three generating sets of the characteristic subalgebras of the symplectic and orthogonal quantum matrix algebras. One of these -- the set of the elementary sums -- is finite. In the symplectic case the elementary sums are in general algebraically independent. On the contrary, in the orthogonal case the elementary sums turn out to be dependent. We obtain a set of quadratic reciprocal relations for these generators. Next, we resolve the reciprocal relations for the quantum orthogonal matrix algebra extended by the inverse of the quantum matrix. As an auxiliary result, we derive the commutation relations between the q-determinant of the quantum orthogonal matrix and the generators of the quantum matrix algebra, that is, the components of the quantum matrix.

math.QA↗
arXiv2025

Plinths and Pedestals

In this paper we explain what are the plinths and the pedestals of the skyscrapers (=plane partitions), and how one can use them in order to count the skyscrapers.

math.CO↗
arXiv2025

The major index (maj) and its Sch\"utzenberger dual

We construct the independent particle representation for the Semistandard Young Tableaux (SsYT) of skew shape $\lambda/\mu.$ The partition function of this particle system gives the generating function of the SsYT of skew shape $\lambda/\mu.$ Thus we obtain a bijective proof of the Stanley formula for the SsYT generating function. To do this we define for every SsYT $T$ its plinth, $\mathsf{p}\left( T\right) ,$ which is a SsYT of the same shape $\lambda/\mu.$ The set of plinths is finite. Our bijection associates to every SsYT $T$ a pair $\left( \mathsf{p}\left( T\right) ,Y\left( T-\mathsf{p}\left( T\right) \right) \right) ,$ where $Y\left( T-\mathsf{p}\left( T\right) \right) $ is the reading Young diagram of the SsYT $\left( T-\mathsf{p}\left( T\right) \right) $. \newline In particular, every Standard Young Tableau (SYT) $P$ has its plinth, $\mathsf{p}\left( P\right) $. The two statistics of SYT-s -- the volume $\left\vert \mathsf{p}\left( P\right) \right\vert $ and $\mathsf{maj}\left( P\right) $ -- are related via the Sch\"{u}tzenberger involution $Sch:$% \[ \left\vert \mathsf{p}\left( P\right) \right\vert =\mathsf{maj}\left( Sch\left( P\right) \right) . \]

math.CO↗
arXiv2024

The miracle of integer eigenvalues

For partially ordered sets $X$ we consider the square matrices $M^{X}$ with rows and columns indexed by linear extensions of the partial order on $X$. Each entry $\left( M^{X}\right)_{PQ}$ is a formal variable defined by a pedestal of the linear order $Q$ with respect to linear order $P$. We show that all the eigenvalues of any such matrix $M^{X}$ are $\mathbb{Z}$-linear combinations of those variables.

math.CO↗
arXiv2022

Art of Unlocking

We describe our recent results concerning the rigidity/unlockability properties of clusters of rigid bodies sliding over the unit sphere.

math.MG↗
arXiv2019

Quantum Matrix Algebras of BMW type: Structure of the Characteristic Subalgebra

A notion of quantum matrix (QM-) algebra generalizes and unifies two famous families of algebras from the theory of quantum groups: the RTT-algebras and the reflection equation (RE-) algebras. These algebras being generated by the components of a `quantum' matrix $M$ possess certain properties which resemble structure theorems of the ordinary matrix theory. It turns out that such structure results are naturally derived in a more general framework of the QM-algebras. In this work we consider a family of Birman-Murakami-Wenzl (BMW) type QM-algebras. These algebras are defined with the use of R-matrix representations of the BMW algebras. Particular series of such algebras include orthogonal and symplectic types RTT- and RE- algebras, as well as their super-partners. For a family of BMW type QM-algebras, we investigate the structure of their `characteristic subalgebras' --- the subalgebras where the coefficients of characteristic polynomials take values. We define three sets of generating elements of the characteristic subalgebra and derive recursive Newton and Wronski relations between them. We also define an associative $\star$-product for the matrix $M$ of generators of the QM-algebra which is a proper generalization of the classical matrix multiplication. We determine the set of all matrix `descendants' of the quantum matrix $M$, and prove the $\star$-commutativity of this set in the BMW type.

math.QA↗
arXiv2019

Critical configurations of solid bodies and the Morse theory of MIN functions

We study the manifold of clusters of nonintersecting congruent solid bodies, all touching the central ball $B\subset\mathbb{R}^{3}$ of radius one. Two main examples are clusters of balls and clusters of infinite cylinders. We introduce the notion of \textit{critical cluster} and we study several critical clusters of balls and of cylinders. For the case of cylinders some of our critical clusters are new. We also establish the criticality properties of clusters, introduced earlier by W. Kuperberg.

math.MG↗
arXiv2019

The Six Cylinders Problem: $\mathbb{D}_{3}$-symmetry Approach

Motivated by a question of W. Kuperberg, we study the 18-dimensional manifold of configurations of 6 non-intersecting infinite cylinders of radius $r,$ all touching the unit ball in $\mathbb{R}^{3}.$ We find a configuration with \[ r=\frac{1}{8}\left( 3+\sqrt{33}\right) \approx1.093070331\ .\] We believe that this value is the maximal possible.

math.MG↗
arXiv2019

Platonic Compounds of Cylinders

Generalizing the octahedral configuration of six congruent cylinders touching the unit sphere, we exhibit configurations of congruent cylinders associated to a pair of dual Platonic bodies.

math.MG↗
arXiv2019

Extremal Cylinder Configurations II: Configuration $O_6$

We study the octahedral configurations $O_6$ of six equal cylinders touching the unit sphere. We show that the configuration $O_6$ is a local sharp maximum of the distance function. Thus it is not unlockable and, moreover, rigid.

math.MG↗
arXiv2018

Extremal Cylinder Configurations I: Configuration $C_{\mathfrak{m}}$

We study the path $Γ=\{ C_{6,x}\ \vert\ x\in [0,1]\}$ in the moduli space of configurations of 6 equal cylinders touching the unit sphere. Among the configurations $C_{6,x}$ is the record configuration $C_{\mathfrak{m}}$ of \cite{OS}. We show that $C_{\mathfrak{m}}$ is a local sharp maximum of the distance function, so in particular the configuration $C_{\mathfrak{m}}$ is not only unlockable but rigid. We show that if $\frac{(1 + x) (1 + 3 x)}{3}$ is a rational number but not a square of a rational number, the configuration $C_{6,x}$ has some hidden symmetries, part of which we explain.

math.MG↗
arXiv2017

Differential Calculus on h-Deformed Spaces

We construct the rings of generalized differential operators on the ${\bf h}$-deformed vector space of ${\bf gl}$-type. In contrast to the $q$-deformed vector space, where the ring of differential operators is unique up to an isomorphism, the general ring of ${\bf h}$-deformed differential operators $\operatorname{Diff}_{{\bf h},σ}(n)$ is labeled by a rational function $σ$ in $n$ variables, satisfying an over-determined system of finite-difference equations. We obtain the general solution of the system and describe some properties of the rings $\operatorname{Diff}_{{\bf h},σ}(n)$.

math.RA↗
arXiv2009

Diagonal reduction algebras of $\gl$ type

Several general properties, concerning reduction algebras - rings of definition and algorithmic efficiency of the set of ordering relations - are discussed. For the reduction algebras, related to the diagonal embedding of the Lie algebra $gl_n$ into $gl_n \oplus gl_n$, we establish a stabilization phenomenon and list the complete sets of defining relations.

math.RT↗
arXiv2007

On rime Ansatz

The ice Ansatz on matrix solutions of the Yang-Baxter equation is weakened to a condition which we call rime. Generic rime solutions of the Yang-Baxter equation are described. We prove that the rime non-unitary (respectively, unitary) R-matrix is equivalent to the Cremmer-Gervais (respectively, boundary Cremmer-Gervais) solution. Generic rime classical r-matices satisfy the (non-)homogeneous associative classical Yang-Baxter equation.

math.QA↗
arXiv2007

Nombres de Bernoulli et une formule de Ramanujan

In the first part we establish a connection between the Euler-Maclaurin summation formula and the Rota-Baxter functional equation. In the second part we give a simple proof of a formula, due to Ramanujan, on the summation of certain exponential series.

math.CA↗