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 [35] 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 relations for these generators. We call these relations ‘reciprocal’ because they lie at the heart of the reciprocal (sometimes called palindromic) property of the characteristic polynomial of the orthogonal quantum matrices. 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.

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 = 2P-1) and even (k = 2P) 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+(2P) and another one for the negative component O-(2P). In each case we introduce the spectral parameterization of the coefficients of the Cayley-Hamilton identity by the 'eigenvalues' of the quantum matrices. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

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.

We construct the independent particle representation for the Semistandard Young Tableaux (SsYT) of skew shape λ/μ. The partition function of this particle system gives the generating function of the SsYT of skew shape λ/μ. 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, 𝗉( T) , which is a SsYT of the same shape λ/μ. 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 Schützenberger involution Sch:

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 Schützenberger involution Sch: |p(P)|=maj(Sch(P)).

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

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.

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.

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.

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.

We establish two versions of the fusion procedure for the walled Brauer algebras. In each of them, a complete system of primitive pairwise orthogonal idempotents for the walled Brauer algebra is constructed by consecutive evaluations of a rational function in several variables on contents of standard walled tableaux.

We construct a normal form for the walled Brauer algebra, together with the reduction algorithm. We apply normal form to calculate the numbers of monomials in generators with minimal length. We further utilize normal form to give explicit expressions for a generating set and annihilator ideal of a particular cyclic vector in a cell module.

Motivated by a question of W. Kuperberg, we study the 18-dimensional manifold of configurations of six non-intersecting infinite cylinders of radius r , all touching the unit ball in ℝ^3 . We find a configuration with r=1/8 ( 3+√(33) ) ≈ 1.093070331. We believe that this value is the maximum possible.

This paper studies the manifold of clusters of non-intersecting congruent solid bodies, all touching the central ball B subset of R-3 of radius one. Two main examples are clusters of balls and clusters of infinite cylinders. The notion of critical cluster is introduced, and several critical clusters of balls and of cylinders are studied. In the case of cylinders, some of the critical clusters here are new. The paper also establishes criticality properties of clusters introduced earlier by Kuperberg [7].

We define contravariant forms on diagonal reduction algebras, algebras of h-deformed differential operators and on standard modules over these algebras. We study properties of these forms and their specializations. We show that the specializations of the forms on the spaces of h-commuting variables present zero singular vectors iff they are in the kernel of the specialized form. As an application we compute norms of highest weight vectors in the tensor product of an irreducible finite dimensional representation of the Lie algebra gln with a symmetric or wedge tensor power of its fundamental representation.

For a linear extension P of a partially ordered set S, we consider a generating multivariate polynomial of certain reverse partitions on S, called P-pedestals. We establish a remarkable property of this polynomial: it does not depend on the choice of P. For S a Young diagram, we show that this polynomial generalizes the hook polynomial.

Analogues of 1-shuffle elements for complex reflection groups of type $G(m,1,n)$ are introduced. A geometric interpretation for $G(m,1,n)$ in terms of rotational permutations of polygonal cards is given. We compute the eigenvalues, and their multiplicities, of the 1-shuffle element in the algebra of the group $G(m,1,n)$. Considering shuffling as a random walk on the group $G(m,1,n)$, we estimate the rate of convergence to randomness of the corresponding Markov chain. We report on the spectrum of the 1-shuffle analogue in the cyclotomic Hecke algebra $H(m,1,n)$ for $m=2$ and small $n$.