Holomorphic Cohomology of Complex Analytic Linear Groups

We shall explain here an idea to generalize classical complex analytic Kleinian group theory to any odd-dimensional cases. For a certain class of discrete subgroups of $\text{PGL}_{2n+1}(\mathbf{C})$ acting on $\mathbf{P}^{2n+1}$, we can define their domains of discontinuity in a canonical manner, regarding an $n$-dimensional projective linear subspace in $\mathbf{P}^{2n+1}$ as a point, like a point in the classical one-dimensional case. Many interesting (compact) non-Kähler manifolds appear systematically as the canonical quotients of the domains. In the last section, we shall give some examples.

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

Let G be a connected semi-simple Lie group with finite center, say — then it is not necessarily the case that G possesses enough finite dimensional representations to separate points (cf. 3.1.1). In fact G will, in general, admit no non-trivial finite dimensional unitary representations (cf. number 4.3.2). Nevertheless, despite these apparently discouraging facts, the finite dimensional representations of G are important — this will be elaborated on below.

Let be a rational finite dimensional complex representation of a reductive linear algebraic group G, and let be a system of generators of the algebra of invariant poly- nomials ℂ[V]G. We study the problem of lifting mappings over the mapping of invariants . Note that ¾(V) can be identified with the cate- gorical quotientV//G and its points correspond bijectively to the closed orbits inV. We prove that if f belongs to a quasianalytic subclass satisfying some mild closedness properties that guarantee resolution of singularities in 𝒞 e.g., the real analytic class, then f admits a lift of the same class 𝒞 after desingularization by local blow-ups and local power substitutions. As a consequence we show that f itself allows for a lift that belongs to SBVloc, i.e., special functions of bounded variation. If ρ is a real representation of a compact Lie group, we obtain stronger versions.

Determinantal method and the Littlewood-Richardson rule

On the cohomology of Coxeter groups and their finite parabolic subgroups. II by T. Akita Linear source modules and trivial source modules by R. Boltje Resolutions de foncteurs de Mackey by S. Bouc Homotopy uniqueness of $BPU(3)$ by C. Broto and A. Viruel Lowering operators for $GL(n)$ and quantum $GL(n)$ by J. Brundan Homomorphisms in higher complexity quotient categories by J. F. Carlson and W. W. Wheeler On the homotopy theory of $p$-completed classifying spaces by F. R. Cohen and R. Levi $H^*V$ is of bounded type over $\mathcal {A} (p)$ by M. D. Crossley Classifying spaces of central group extensions by J. Dietz and T. Ratliff Sharp homology decompositions for classifying spaces of finite groups by W. G. Dwyer On the geometry and cohomology of the simple groups $G_2(q)$ and $^3D_4(q)$ by P. Fong and R. J. Milgram Transfer and Chern classes for extraspecial $p$-groups by D. J. Green and P. A. Minh The infinite order of the symplectic classes in the cohomology of the stable mapping class group by R. Grieder A variant of the proof of the Landweber Stong conjecture by H.-W. Henn Unstable modules over the Steenrod algebra and cohomology of groups by H.-W. Henn The category of Mackey functors for a compact Lie group by L. G. Lewis, Jr. Comparison of extensions of modules for algebraic groups and quantum groups by Z. Lin The defect of the completion of a Green functor by F. Luca Simple modules of Ariki-Koike algebras by A. Mathas On the geometry and cohomology of the simple groups $G_2(q)$ and $^3D_4(q)$: II by R. J. Milgram On the relation between simple groups and homotopy theory by R. J. Milgram Varieties for $G_rT$-modules by D. K. Nakano On defect groups of the Mackey algebras for finite groups by F. Oda Applications of stable classifying space theory to group cohomology by S. Priddy The spectra $k$ and $kO$ are not Thom spectra by Yu. B. Rudyak The code of a regular generalized quadrangle of even order by N. S. N. Sastry and P. Sin Quivers of blocks with normal defect group by M. Schaps, D. Shapira, and O. Shlomo Cohomology of a homogeneous space $E_6/T^1\cdot SU(6)$ by T. Watanabe Problem session, Seattle 1996 by D. Benson.

Introduction. In applying symmetry considerations to quantum mechanics, one is often forced to consider representations T of a symmetry group G by means of Hilbert space operators which may be either unitary or conjugate-unitary. Indeed, in some situations there is a well-determined subgroup N of G (necessarily normal and of index 2 in G) such that T, must be unitary for x E N and conjugate-unitary for x E G N. Such a representation T will be called a conjugating representation of G (relative to N). More generally one can consider projective conjugating representations T, in which the homomorphism relation TT, = T, is replaced by Tj, = A(x, y)Txy (the A(x, y) being complex scalars). These also are required in the applications to quantum mechanics. In ?1, by a small modification of Mackey's analysis, we shall classify the equivalence classes of irreducible projective conjugating representations of G in terms of the irreducible projective representations of N (provided N is of Type I). As an immediate application of this, we obtain in ?2 a classification of all the irreducible unitary representations of any Type I group acting in real or quaternionic Hilbert space, in terms of those acting in complex Hilbert space. In working out this classification for a given group G, the essential step is to subdivide the self-conjugate irreducible complex representations T of G into two classes-those of real type and those of quaternionic type-according as the conjugate-linear map setting up the equivalence of T with its conjugate has positive or negative square. Exactly the same results are obtained if, instead of unitary (not necessarily finite-dimensional) representations, we consider finite-dimensional (not necessarily unitary) representations. The remaining sections of this paper concern finite-dimensional representations of semisimple groups. Fix a connected semisimple Lie group G. The family G of all finite-dimensional (not necessarily unitary) irreducible complex representations of G is parametrized by the well-known Cartan-Weyl method of dominant weights. Which of the self-conjugate elements of Gi are of real type and which are of quaternionic type? In ??3 and 4 this question is answered in two steps: First, Theorem 5 reduces the general case to the case that G is compact. It turns out that, if G is not compact, each self-conjugate element T of G gives rise to a self-conjugate element

Let Ik denote the identity matrix of order k and set **-Let SrX&,) denote the group of all complex Ik X k matrices which satisfy the equation gskg' sk.Let be the linear space of all n X 2k complex matrices with k > n, and let S(E*) denote the symmetric algebra of all complex-valued polynomial functions on E. The study of the action of Sp(At,C), which is obtained by right translation on S(E*), leads to a concrete and simple realization of all irreducible holomorphic representations of Sp(k,C).In connection with this realization, a theory of symplectic Stiefel harmonics is also established.This notion may be thought of as a generalization of the spherical harmonics for the symplectic Stiefel manifold. Introduction. Let Ik denote the unit matrix of order k and setLet Sp(zt,C) denote the group of all complex 2k X 2k matrices g satisfying gskg' = sk.If G is a complex Lie group and F is a finite-dimensional complex vector space, then by a holomorphic representation of G in V we mean a complex analytic homomorphism of G into GL(F).The object of this paper is to present, in as simple and explicit a form as possible, a description of all irreducible holomorphic representations (up to equivalence) of the complex linear symplectic group Sp(,C).This realization is obtained by analyzing certain natural actions of Sp(:,C) on a symmetric algebra of polynomial functions of matrix argument.We rely on some general results of S. Helgason [3] and B. Kostant [4] to study this symmetric algebra.The techniques they used to decompose a symmetric algebra of polynomial functions were based on the existence of a symmetric bilinear form defined on this algebra.The situation is somewhat different in the context of our investigation, because

As the structure and behavior of molecules and crystals depend on their different symmetries, group theory becomes an essential tool in many important areas of chemistry. It is a quite powerful theoretical tool to predict many basic as well as some characteristic properties of molecules. Whereas quantum mechanics provide solutions of some chemical problems on the basis of complicated mathematics, group theory puts forward these solutions in a very simplified and fascinating manner. Group theory has been successfully applied to many chemical problems. Students and teachers of chemical sciences have an invisible fear from this subject due to the difficulty with the mathematical jugglery. An active sixth dimension is required to understand the concept as well as to apply it to solve the problems of chemistry. This book avoids mathematical complications and presents group theory so that it is accessible to students as well as faculty and researchers. Chemical Applications of Symmetry and Group Theory discusses different applications to chemical problems with suitable examples. The book develops the concept of symmetry and group theory, representation of group, its applications to I.R. and Raman spectroscopy, U.V spectroscopy, bonding theories like molecular orbital theory, ligand field theory, hybridization, and more. Figures are included so that reader can visualize the symmetry, symmetry elements, and operations.

Wreath products (defined below) seem to be ubiquitous in group theory and its applications. They appear implicitly in the work of 19th century group theorists, but were first defined explicitly by Polya when studying combinatorial problems associated with classifying organic molecules (see Polya and Read 1987, p.99). They were studied systematically in a series of papers by Kaloujnine and Krasner (1948, 1950, 1951a, 1951b). The Sylow subgroups of symmetric groups (Hall 1959, Section 5.9) are formed from wreath products, as are the Sylow subgroups of Gl(n, Fq) (and other linear groups) away from the characteristic of the field (Weir 1955). The cohomology of wreath products has played an important role in the development of the subject, in part because of interest in the cohomology of such groups, but also because wreath products are closely connected with the notion of transfer (corestriction) and its generalizations. We shall explore these notions in what follows.

Let Γ be a connected reductive group over an algebraically closed field F of positive characteristic and let G be the set of fixed points of Γ under some Frobenius map of Γ. Then G is a finite group of Lie type. The p-modular representations of G for p = char F (the “describing” characteristic case) are closely related to rational representations of Γ, and thus results from the theory of reductive algebraic groups can be used to develop the representation theory of finite groups of Lie type in the describing characteristic case. One of the outstanding open questions in this area is to determine the dimensions of the irreducible representations and the decomposition matrix whose entries record multiplicities of irreducible modules as composition factors of Weyl modules. One way to define Weyl modules is by reducing modulo p (p = char F) the irreducible modules of the complex algebraic group of the same type. Here reduction modulo p happens in both, in the field underlying the group and in the field over which the representing matrices are defined, simultaneously. One of the main tools to study these questions is derived from the Weyl group of Γ. If, for example, Γ is a general linear group, the decomposition matrix of the Weyl modules can be calculated from decomposition numbers of Schur algebras, which in turn can be calculated in terms of symmetric groups. We shall discuss this phenomenon in more detail below.

The asymptotic root distribution is computed for systems of matrix functions associated with finite-dimensional holomorphic representations of a Lie group. This distribution can be expressed via the increments of the representations involved. If the group is reductive, then the number of equations in the system can be arbitrary, from 1 to the dimension of the group. In this case, the computation results are stated in the language of convex geometry. These computations imply the previously known formulas for the density of the solution variety of a system of exponential equations as well as for the number of solutions of a polynomial system and, more generally, of a system formed by matrix functions of representations of a complex reductive Lie group.

The cohomology of groups

The work is devoted to the consideration of applications of mathematics, in particular, graph theory and group theory in chemistry and physics. Chemistry was one of the first to adopt the ideas of graph theory. Mathematical chemistry is a section of theoretical chemistry that studies the application of mathematics to chemical problems. The possibility of formalizing molecules in the language of structural chemistry using graphs allows one to determine the number of different isomers using graph theory methods. The main tool in mathematical chemistry is mathematical modeling using computer technology. The methods of graph theory are actively used in three areas of chemistry: structural chemistry, chemical kinetics, and chemical physics of polymers. Group theory has found application in many areas of physics. Group theory serves to explain the most important characteristics of atomic spectra. Group theory provides an opportunity for a uniform approach to a large number of problems in solid state physics. Matrix groups and unitary group representations are actively used in these studies. Irreducible group representations are used to classify vibrational and electronic states of a crystal. Methods of group theory are applied to models of problems of classifying electronic states of a crystal. In crystallography, much attention is paid to applications of group theory methods to the study of vibrations of atoms that make up a molecule relative to their equilibrium positions. The results can be useful in encoding information in space communications sessions.

Let G be a split semisimple algebraic group over 𝐐 with trivial center. Let S be a compact oriented surface, with or without boundary. We define positive representations of the fundamental group of S to G ( 𝐑 ) , construct explicitly all positive representations, and prove that they are faithful, discrete, and positive hyperbolic; the moduli space of positive representations is a topologically trivial open domain in the space of all representations. When S have holes, we defined two moduli spaces closely related to the moduli spaces of G-local systems on S. We show that they carry a lot of interesting structures. In particular we define a distinguished collection of coordinate systems, equivariant under the action of the mapping class group of S. We prove that their transition functions are subtraction free. Thus we have positive structures on these moduli spaces. Therefore we can take their points with values in any positive semifield. Their positive real points provide the two higher Teichmüller spaces related to G and S, while the points with values in the tropical semifields provide the lamination spaces. We define the motivic avatar of the Weil-Petersson form for one of these spaces. It is related to the motivic dilogarithm.