A Constructive Proof of Brauer's Theorem on Induced Characters in the Group Ring
We provide an alternative constructive proof of the classical Brauer theorem for finite groups based on the well-known description of the complex irreducible representations of the symmetric groups Sn . The theorem is first proved for Sn and then for general G by embedding in Sn and applying the Mackey subgroup theorem.
The application of the evolutionary approach to the history of nature and society has remained one of the most effective ways to conceptualize and integrate our growing knowledge of the Universe, life, society and human thought. The present volume demonstrates this in a rather convincing way. This is the third issue of the Almanac series titled ‘Evolution’. The first volume came out with the sub-heading ‘Cosmic, Biological, and Social’, the second was entitled ‘Evolution: A Big History Perspective’. The present volume is subtitled Development within Big History, Evolutionary and World-System Paradigms. In addition to the straightforward evolutionary approach, it also reflects such adjacent approaches as Big History, the world-system analysis, as well as globalization paradigm and long wave theory. The volume includes a number of the exciting works in these fields.
The Almanac consists of five sections. The first section (Globalization as an Evolutionary Process: Yesterday and Today) contains articles demonstrating that the Evolutionary studies is capable of creating a common platform for the world-system approach, globalization studies, and the economic long-wave theory. The articles of the second section (Society, Energy, and Future) discuss the role of energy in the universal evolution, human history and the future of humankind. The third section (Aspects of Social Development) touches upon four aspects of social evolution – technological, environmental, cultural, and political. The fourth section (The Driving Forces and Patterns of Evolution) deals with various phases of megaevolution. There is also a final section which is devoted to discussions of contemporary evolutionism.
This Almanac will be useful both for those who study interdisciplinary macroproblems and for specialists working in focused directions, as well as for those who are interested in evolutionary issues of Cosmology, Biology, History, Anthropology, Economics and other areas of study. More than that, this edition will challenge and excite your vision of your own life and the new discoveries going on around us!
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.