The Schlesinger system and isomonodromic deformations of bundles with connections on Riemann surfaces

We introduce a way of presentation of pairs $(E,\nabla)$, where $E$ is a bundle on a Riemann surface and $\nabla$ is a logarithmic connection in $E$, which is based on a presentation of the surface as a factor of the exterior of the unit disc. In this presentation we write the local equation of isomonodormic deformation of pairs $(E,\nabla)$. These conditions are written as a modified Schlesinger system on a Riemann sphere (and in the typical case just as an ordinary Schlesinger system) plus some linear system.Comment: 19 pages, final version accepted to publicatio

Models of representations for classical series of Lie algebras

A model of representations of a Lie algebra is a representation which a direct sum of all irreducible finite dimensional representations taken with multiplicity $1$. In the paper an explicit construction of a model of representation for all series of classical Lie algebras is given. The construction does not differ much for different series. The space of the model is constructed as a space of polynomial solutions of a system of partial differential equations. The equations in this system are constructed form relations between minors of matrices from the corresponding Lie group. This system has a simplification which is very close to the GKZ system, that is satisfied by $A$-hypergeometric functions.Comment: 41 page

Classical 6j6j-symbols for finite dimensional representation of the algebra gl3\mathfrak{gl}_3

In the paper an explicit formula for an arbitrary $6j$-symbol for finite-dimensional irreducible representations of the algebra $\mathfrak{gl}_3$ is derived. A $6j$-symbol is written as a result of substitution of $\pm 1$ into a series of hypergeometric type, which is similar to a $\Gamma$-series, which is a simplest example of a multivariate series of hypergeometric type. The selection rulers for a $6j$-symbol are derived.Comment: 21 page