Klein Groups Research Articles

Poincaré series of Klein groups, coxeter polynomials, the Burau representation, and Milnor invariants

We obtain several formulas for the Poincare series defined by B. Kostant for Klein groups (binary polyhedral groups) and some formulas for Coxeter polynomials (characteristic polynomials of monodromy in the case of singularities). Some of these formulas—the generalized Ebeling formula, the Christoffel-Darboux identity, and the combinatorial formula—are corollaries to the well-known statements on the characteristic polynomial of a graph and are analogous to formulas for orthogonal polynomials. The ratios of Poincare series and Coxeter polynomials are represented in terms of branched continued fractions, which are q-analogs of continued fractions that arise in the theory of resolution of singularities and in the Kirby calculus. Other formulas connect the ratios of some Poincare series and Coxeter polynomials with the Burau representation and Milnor invariants of string links. The results obtained by S.M. Gusein-Zade, F. Delgado, and A. Campillo allow one to consider these facts as statements on the Poincare series of the rings of functions on the singularities of curves, which suggests the following conjecture: the ratio of the Poincare series of the rings of functions for close (in the sense of adjacency or position in a series) singularities of curves is determined by the Burau representation or by the Milnor invariants of a string link, which is an intermediate object in the transformation of the knot of one singularity into the knot of the other.

Orthogonal Cayley-Klein Groups

A. Cayley [5] and F. Klein [12] were the first mathematicians who introduced the notion of a metric in a real projective space, by specializing a set of quadrics (called the absolute). We show in this paper, that from an algebraic point of view these projective spaces can be described by vector spaces, in which a metric structure is given by one or more symmetric bilinear forms. We study these Cayley- Klein vector spaces and their groups of automorphisms (orthogonal Cayley- Klein groups) over arbitrary commutative fields of characteristic ≠ 2.

Survival and Proliferation Factors of Normal and Malignant Plasma Cells

Since the first identification of interleukin (IL)-6 as a myeloma cell growth factor by Dr. Kawano's and Dr. Klein's groups 14 years ago, numerous studies have emphasized its major roles in the emergence of malignant plasma cells in vivo and in the generation of normal plasma cells. Four transcription factors control B-cell differentiation into plasma cells. The B-cell transcription factor pax-5 is mainly responsible for a B-cell phenotype, and bcl-6 represses the plasma cell transcription factor blimp-1 and plasma cell differentiation. bcl-6 expression is triggered by CD40 and IL-4 activation. A lack of CD40 and IL-4 activation yields a down-regulation of bcl-6 expression, and IL-6 stimulation yields an up-regulation of blimp-1, mainly through STAT3 activation. Blimp-1 further down-regulates bcl-6 and pax-5 expression and makes plasma cell differentiation possible. IL-6 as well as IL-10 up-regulate XBP-1. XBP-1 is another transcription factor that is involved in plasma cell differentiation and whose gene expression is shut down by pax-5. The plasma cell transcription factors blimp-1 and XBP-1 are up-regulated, and the B-cell transcription factors bcl-6 and pax-5 are down-regulated, in malignant cells compared to B-cells. Apart from the recent identification of these 4 transcription factors, the factors involved in normal plasma cell generation are mostly unknown. Regarding malignant plasma cells, 3 categories of growth factors have been identified: (1) the IL-6 family cytokines, IL-10, and interferon alpha that activate the Janus kinase-signal transducer and activator of transcription (JAK/STAT) and mitogen-activated protein (MAP) kinase pathways; (2) growth factors activating the phosphatidylinositol (PI)-3 kinase/AKT and MAP kinase pathways, unlike the JAK/STAT pathway (insulin-like growth factor 1, hepatocyte growth factor, and members of the epidermal growth factor family able to bind syndecan-1 proteoglycan); and (3) B-cell-activating factor (BAFF) or proliferation-inducing ligand (APRIL) that activate the nuclear factor KB and PI-3 kinase/AKT pathways. BAFF and APRIL bind to BAFF receptor and TACI and are major B-cell survival factors. Recent data indicate that these various growth factors may cooperate to provide optimum signaling because they are localized together and with cytoplasmic transduction elements in caveolinlinked membrane caveolae. The identification of these myeloma cell growth factors and of the associated transduction pathways should provide novel therapeutic targets in multiple myeloma.

The Jordan–Schwinger representations of Cayley–Klein groups. I. The orthogonal groups

The Cayley–Klein groups are defined as the groups that are obtained by the contractions and analytical continuations of the orthogonal groups. The Jordan–Schwinger representations of Cayley–Klein groups are discussed based on the mixed sets of creation and annihilation operators of boson or fermion type. The matrix elements of finite group transformations are obtained in the bases of coherent and Fock states.

The Jordan–Schwinger representations of Cayley–Klein groups. III. The symplectic groups

The symplectic Cayley–Klein groups are defined as the groups that are obtained by the contractions and analytical continuations of the classical symplectic groups. The Jordan–Schwinger representations of the groups under consideration are discussed based on the mixed sets of creation and annihilation operators of boson or fermion type. The matrix elements of finite group transformations are obtained in the bases of coherent states.

The Jordan–Schwinger representations of Cayley–Klein groups. II. The unitary groups

The unitary Cayley–Klein groups are defined as the groups that are obtained by the contractions and analytical continuations of the special unitary groups. The Jordan–Schwinger representations of the groups under consideration are discussed based on the mixed sets of creation and annihilation operators of boson or fermion type. The matrix elements of finite group transformations are obtained in the bases of coherent states.

Klein groups in a space

Klein groups in a space

Finite rotation groups and crystal classes in four dimensions

The groups of symmetries of three-dimensional lattices have been known for some time. They consist of finite rotation groups, the crystal classes, and infinite discrete motion groups, which include both rotations and translations. The general theory of the corresponding groups in higher dimensional Euclidean spaces has also been developed. This theory includes a demonstration that in Euclidean space of n dimensions the number of motion groups is finite, and leads to a method† for calculating the motion groups, the first step being to determine the crystal classes. The explicit calculation of the various groups by the general method is not simple, and has so far been confined to the case of two and three dimensions. In the special case of the crystal classes in four dimensions, however, we may make use of the results of a paper by Goursat‡. In this paper Goursat sets up a correspondence between the finite rotation groups in four-dimensional Euclidean space, and a set of groups each of which is formed by associating two of Klein's groups of linear non-homogeneous substitutions in one variable. Using this result he is able to evaluate explicitly all the proper and improper finite four-dimensional rotation groups which include the element −I, where I is the four-rowed unit matrix.