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We give a bijective proof of the Stanley formula for the generating function of the Semistandard Young Tableaux (SsYT) of skew shape {\lambda}/{\mu}. To do this we define for every SsYT T its plinth, p(T), 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 (p(T),Y(T-p(T))), where Y(T-p(T)) is the reading Young diagram of the SsYT T-p(T). In particular, every Standard Young Tableau (SYT) P has its plinth, p(P). The two statistics of SYT-s -- the volume |p(P)| and maj(P) -- are related via the Sch\"utzenberger involution Sch: |p(P)|=maj(Sch(P))., Comment: 18 pages

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.

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

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., Comment: arXiv admin note: substantial text overlap with arXiv:math/0511618

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., Comment: arXiv admin note: text overlap with arXiv:1812.09543

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.

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.

We study the path $\Gamma=\{ C_{6,x}\ \vert\ x\in [0,1]\}$ 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.

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.

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 $\operatorname{Diff}_{{\bf h},\sigma}(n)$ is labeled by a rational function $\sigma$ 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 $\operatorname{Diff}_{{\bf h},\sigma}(n)$.

We define, for an arbitrary partially ordered set, a multi-variable polynomial generalizing the hook polynomial.

A complete system of primitive pairwise orthogonal idempotents for cyclotomic Hecke algebras is constructed by consecutive evaluations of a rational function in several variables on quantum contents of multi-tableaux. This function is a product of two terms, one of which depends only on the shape of the multi-tableau and is proportional to the inverse of the corresponding Schur element.

Quantum Lie algebras related to multi-parametric Drinfeld-Jimbo $R$-matrices of type $GL(m|n)$ are classified., Comment: 9 pages

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., Comment: 24 pages, no figures

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., Comment: 4 pages, talk given at the VII International Workshop "Supersymmetries and Quantum Symmetries", Dubna 2007

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., Comment: 27 pages

For families of orthogonal and symplectic types quantum matrix (QM-) algebras, we derive corresponding versions of the Cayley-Hamilton theorem. For a wider family of Birman-Murakami-Wenzl type QM-algebras, we investigate a structure of its characteristic subalgebra (the subalgebra in which the coefficients of characteristic polynomials take values). We define 3 sets of generating elements of the characteristic subalgebra and derive recursive Newton and Wronski relations between them. For the family of the orthogonal type QM-algebras, additional reciprocal relations for the generators of the characteristic subalgebra are obtained., Comment: 69 pages

We give the inflation rules for the decorated Mosseri-Sadoc tiles in the projection class of tilings ${\cal T}^{(MS)}$. Dehn invariants related to the stone inflation of the Mosseri-Sadoc tiles provide eigenvectors of the inflation matrix with eigenvalues equal to $\tau = \frac{1+\sqrt{5}}{2}$ and $-\tau^{-1}$., Comment: LaTeX file, 4(3) pages + 7 figures (FIG1.gif, FIG2.gif,... FIH7.gif) and a style file (icqproc.sty)

We introduce Dehn invariants as a useful tool in the study of the inflation of quasiperiodic space tilings. The tilings by ``golden tetrahedra'' are considered. We discuss how the Dehn invariants can be applied to the study of inflation properties of the six golden tetrahedra. We also use geometry of the faces of the golden tetrahedra to analyze their inflation properties. We give the inflation rules for decorated Mosseri-Sadoc tiles in the projection class of tilings ${\cal T}^{(MS)}$. The Dehn invariants of the Mosseri-Sadoc tiles provide two eigenvectors of the inflation matrix with eigenvalues equal to $\tau = \frac{1+\sqrt{5}}{2}$ and $-\frac{1}{\tau}$, and allow to reconstruct the inflation matrix uniquely., Comment: LaTeX file, 25 pages + 9 figures (Fig1.gif, Fig2.gif... Fig9.gif); hard copies with all figures are available from the authors

We define quantum matrix groups GL(3) by their coaction on appropriate quantum planes and the requirement that the Poincare series coincides with the classical one. It is shown that this implies the existence of a Yang-Baxter operator. Exploiting stronger equations arising at degree four of the algebra, we classify all quantum matrix groups GL(3). We find 26 classes of solutions, two of which do not admit a normal ordering. The corresponding R-matrices are given., Comment: 28 pages, Latex

We construct for all $N$ a solution of the Frenkel--Moore $N$--simplex equation which generalizes the $R$--matrix for the Jordanian quantum group., Comment: 6 pages

The expressions for the $\hat{R}$--matrices for the quantum groups SO$_{q^2}$(5) and SO$_q$(6) in terms of the $\hat{R}$--matrices for Sp$_q$(2) and SL$_q$(4) are found, and the local isomorphisms of the corresponding quantum groups are established., Comment: 11 pages