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## Branching rules related to spherical actions on flag varieties

Let G be a connected semisimple algebraic group and let H⊂G be a connected reductive subgroup. Given a flag variety X of G, a result of Vinberg and Kimelfeld asserts that H acts spherically on X if and only if for every irreducible representation R of G realized in the space of sections of a homogeneous line bundle on X the restriction of R to H is multiplicity free. In this case, the information on restrictions to H of all such irreducible representations of G is encoded in a monoid, which we call the restricted branching monoid. In this paper, we review the cases of spherical actions on flag varieties of simple groups for which the restricted branching monoids are known (this includes the case where H is a Levi subgroup of G) and compute the restricted branching monoids for all spherical actions on flag varieties that correspond to triples (G,H,X) satisfying one of the following two conditions: (1) G is simple and H is a symmetric subgroup of G; (2) G=SL_n.

Formation of democratic societies of the Western type presupposes appearance on the historical scene of a new strong actor - the bourgeois class: "No bourgeoisie, no democracy" (Barrington Moore). The articulation and defense of vital interests of that class creates a new social space - "the bourgeois public sphere" which helps to make up "counterbalance" to absolutism of a corporate state - a civil society, the core of which is composed by public opinion. In the confrontation between the authorities and society one of the most important roles is played by the press that provides free debate and discussion of generally valid problems, especially economic and political. The recognition of the mass media role was stamped in its characterization in XIII century as "the fourth power". Technological development of the media incredibly expanded its functions, turning journalists into creating informational analogue of reality, saturating daily life with new meanings. Methods of the representation of reality, the specific nature of political influence of journalists - key members of the reflexive elites (Helmut Shelski), are the themes of this article.

Публичная сфера, журналистика, четвертая власть, порядки знания, Повседневность, научное и повседневное знание, экспертиза, Репрезентация, public sphere, journalism, fourth estate, orders of knowledge, Everyday life, scientific and everyday knowledge, Expertise, representation

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.

The following topics about subgroups of the Cremona groups are discussed: (1) maximal tori; (2) conjugacy and classification of diagonalizable subgroups of codimensions 0 and 1; (3) conjugacy of finite abelian subgroups; (4) algebraicity of normalizers of diagonalizable subgroups; (5) torsion primes.

The aim of these notes is to give an introduction into Schubert calculus on Grassmannians and flag varieties. We discuss various aspects of Schubert calculus, such as applications to enumerative geometry, structure of the cohomology rings of Grassmannians and flag varieties, Schur and Schubert polynomials. We conclude with a survey of results of V. Kiritchenko, V. Timorin and the author on a new approach to Schubert calculus on full flag varieties via combinatorics of Gelfand-Zetlin polytopes.

Th e article is devoted to the secondary nomination. The essence of the act of nomination is to fi x the communication of the subject and name, the phenomenon and its designation, the structures of the consciousness and its object. Man picks the right means of nomination when forms a notion of an object or phenomenon. The results showed that one of the types of secondary nomination is a semantic transposition, which does not change the material appearance of rethoughtful unit and leads to formation of its new value, i.e. for multiple purposes, namely, to metaphor, in particular, an anthropomorphic comparison.

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.

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.