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## Quantum toroidal gl(1)-algebra: Plane partitions

In this, the third paper of the series, we construct a large family of representations of the quantum toroidal gl(1)-algebra whose bases are parameterized by plane partitions with various boundary conditions and restrictions. We study the corresponding formal characters. As an application We obtain a Gelfand-Zetlin-type basis for a class of irreducible lowest weight gl(infinity)-modules.

We define and study representations of quantum toroidal gln with natural bases labeled by plane partitions with various conditions. As an application, we give an explicit description of a family of highest weight representations of quantum affine gln with generic level.

In this article authors attempted to discover (identify and describe) changes of the symbolical meanings/marks/codes of political transformations within visual representations on the basis of Russian fashion magazines materials covering period between 1980 till 2013 years. On the example of 918 magazines’ and 135 Mersedes-benz Fashion Week Russia 2012/2013 photos, this research completed in screening design of studying processes of political changes with “steps” connected to significant stages of regime transformations. Authors concludes that through fashion magazines, intensively filled by visual tokens, public political processes are not just reflected which allows to clarify their semantically-communicative codes, rather is ongoing anticipatory legitimization of social order and political regime changes, probably initiated and/or supported from powered groups.

Cinematic representations not only strongly influence our interpretation of history (Ferro 1992: 315), but are also important for understanding key aspects of Soviet disability policy. At the beginning of the twentieth century the new medium of cinema enjoyed immense popularity in many countries due to the efforts of commercial filmmakers to produce popular entertainment in the genres of melodrama, comedies and adventure stories. After the October Revolution in Russia, however, cinema was mainly used for education and propaganda (Lawton 1992: 2). Visual arts not only represented, but also contributed to, political discourses in Soviet society by using old and new imaginaries for classifying citizens. This chapter explores the ‘iconography’ of disability in Soviet film in order to reveal the shifting and contested meanings associated with the visual representation of disabled bodies

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.