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## Representations of quantum toroidal gln

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, 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 consider partial sum rules for the homogeneous limit of the solution of the q-deformed Knizhnik–Zamolodchikov equation with reflecting boundaries in the Dyck path representation of the Temperley–Lieb algebra. We show that these partial sums arise in a solution of the discrete Hirota equation, and prove that they are the generating functions of τ 2-weighted punctured cyclically symmetric transpose complement plane partitions where τ =−(q+q−1). In the cases of no or minimal punctures, we prove that these generating functions coincide with τ 2-enumerations of vertically symmetric alternating sign matrices and modifications thereof.

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

*Gendering Postsocialism* explores changes in gendered norms and expectations in Eastern Europe and Eurasia after the fall of the Berlin Wall. The dismantlement of state socialism in these regions triggered monumental shifts in their economic landscape, the involvement of their welfare states in social citizenship and, crucially, their established gender norms and relations, all contributing to the formation of the post-socialist citizen. Case studies examine a wide range of issues across 15 countries of the post-soviet era. These include gender aspects of the developments in education in Kazakhstan, Uzbekistan and Hungary, controversies around abortion legislation in Poland, migrant women and housing as a gendered problem in Russia, challenges facing women’s NGOs in Bosnia, and identity formation of unemployed men in Lithuania. This close analysis reveals how different variations of neoliberal ideology, centred around the notion of the self-reliant and self-determining individual, have strongly influenced post-socialist gender identities, whilst simultaneously showing significant trends for a "re-traditionalising" of gender norms and expectations. This volume suggests that despite integration with global political and free market systems, the post-socialist gendered subject combines strategies from the past with those from contemporary ideologies to navigate new multifaceted injustices around gender in Eastern Europe and Eurasia.

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.