Vol 213, No 5 (2016)

A stratification of the manifold of all square matrices is considered. One equivalence class consists of the matrices with the same sets of values of rank(A − λ_iI)^j . The stratification is consistent with a fibration on submanifolds of matrices similar to each other, i.e., with the adjoint orbits fibration. Internal structures of matrices from one equivalence class are very similar; among other factors, their (co)adjoint orbits are birationally symplectomorphic. The Young tableaux technique developed in the paper describes this stratification and the fibration of the strata on (co)adjoint orbits.

We study the quadratic form of the Laplace operator in 3 dimensions written in spherical coordinates and acting on transverse components of vector-functions. Operators which act on parametrizing functions of one of the transverse components with angular momentum 1 and 2 appear to be fourth-order symmetric operators with deficiency indices (1, 1). We consider self-adjoint extensions of these operators and propose the corresponding extensions for the initial quadratic form. The relevant scalar product for angular momentum 2 differs from the original product in the space of vector-functions, but, nevertheless, it is still local in radial variable. Eigenfunctions of the operator extensions in question can be treated as stable soliton-like solutions of the corresponding dynamical system whose quadratic form is a functional of the potential energy.

Algebraic solutions of the sixth Painlevé equation can be constructed with the help of RS-transformations of hypergeometric equations. Construction of these transformations includes specially ramified rational coverings of the Riemann sphere and the corresponding Schlesinger transformations (S-transformations). Some algebraic solutions can be constructed from rational coverings alone, without obtaining the corresponding pullbacked isomonodromy Fuchsian system, i.e., without the S part of the RS transformations. At the same time, one and the same covering can be used to pullback different hypergeometric equations, resulting in different algebraic Painlevé VI solutions. In the case of high degree coverings, construction of the S parts of the RS-transformations may represent some computational difficulties. This paper presents computation of explicit RS pullback transformations and derivation of algebraic Painlevé VI solutions from them. As an example, we present a computation of all seed solutions for pullbacks of hyperbolic hypergeometric equations. Bibliography: 26 titles.

The aim of this paper is to provide an overview of results about classification of quantum groups which were obtained by the authors. Bibliography: 17 titles.

We consider the five-vertex model on an M ×2N lattice with fixed boundary conditions of special type. We discuss a determinantal formula for the partition function in application to description of various enumerations of N × N × (M − N) boxed plane partitions. It is shown that at the free-fermion point of the model, this formula reproduces the MacMahon formula for the number of boxed plane partitions, while for generic weights (outside the free-fermion point), it describes enumerations with the weight depending on the cumulative number of jumps along vertical (or horizontal) rows. Various representations for the partition function which describes such enumerations are obtained.

In this paper, we obtain generating functions of three-variable Chebyshev polynomials (of the first as well as of the second type) associated with the root system of the A₃ Lie algebra. Bibliography: 21 titles.