Coxeter-Dynkin Diagram Research Articles

Chudnovsky-Ramanujan type formulae for non-compact arithmetic triangle groups

We develop a uniform method to derive Chudnovsky-Ramanujan type formulae for triangle groups based on a generalization of a method of Chudnovsky and Chudnovsky; in particular, we carry out the method systematically for non-compact arithmetic triangle groups and one non-Fuchsian covering. As a result, we derive all rational Ramanujan type series given by Chan-Cooper for levels 1-4, as well as two additional rational series of a similar form prescribed by Chan-Cooper for these levels, but not found in the paper of Chan-Cooper. These two additional series were first found by Z.-W. Sun in a slightly different form. We also derive additional rational series of a similar form, but not found in the papers of Chan-Cooper nor Z.-W. Sun.As an ingredient in the method, we give an algorithm to rigorously confirm the singular values of normalized Eisenstein series of weight 2 and give examples in higher class number, which may be of independent interest. We also prove some partial completeness results for Clausen type single summation series.

A new complex reflection group in PU(9,1) and the Barnes–Wall lattice

We show that the projectivized complex reflection group $\Gamma$ of the unique $(1+i)$-modular Hermitian $\mathbb{Z}[i]$-module of signature $(9,1)$ is a new arithmetic reflection group in $PU(9,1)$. We find $32$ complex reflections of order four generating $\Gamma$. The mirrors of these $32$ reflections form the vertices of a sort of Coxeter-Dynkin diagram $D$ for $\Gamma$ that encode Coxeter-type generators and relations for $\Gamma$. The vertices of $D$ can be indexed by sixteen points and sixteen affine hyperplanes in $\mathbb{F}_2^4$. The edges of $D$ are determined by the finite geometry of these points and hyperplanes. The group of automorphisms of the diagram $D$ is $2^4 \colon (2^3 \colon L_3(2)) \colon 2$. This group transitively permutes the $32$ mirrors of generating reflections and fixes an unique point $\tau$ in $\mathbb{C} H^9$. These $32$ mirrors are precisely the mirrors closest to $\tau$. These results are strikingly similar to the results satisfied by the complex hyperbolic reflection group at the center of Allcock's monstrous proposal.

The Mutation Game, Coxeter–Dynkin Graphs, and Generalized Root Systems

We introduce the mutation game on a directed multigraph, which is dual to Mozes’ numbers game. This new game allows us to create geometric and combinatorial structure that allows generalization of root systems to more general graphs. We interpret Coxeter–Dynkin diagrams in this multigraph context and exhibit new geometric forms for the associated root systems.

The polytopes of the H3 group with 60 vertices and their orbit decompositions.

The goal of this article is to compare the geometrical structure of polytopes with 60 vertices, generated by the finite Coxeter group H3, i.e. an icosahedral group in three dimensions. The method of decorating a Coxeter-Dynkin diagram is used to easily read the structure of the reflection-generated polytopes. The decomposition of the vertices of the polytopes into a sum of orbits of subgroups of H3 is given and presented as a `pancake structure'.

The decoration of a Coxeter—Dynkin diagram and the Schläfli symbol as two methods to describe polytopes generated by finite reflection groups

The aim of this paper is to present two methods of describing uniform convex polytopes in three dimensions, namely, the Schläfli symbol and the method of decorating a Coxeter–Dynkin diagram. The reflection-generated polytopes are important objects, both in an abstract and in a physical context. In mathematics such polytopes are orbits of finite reflections groups, while in chemistry, biology and physics they serve as physical models for molecules, proteins, viruses and other three-dimensional structures. For this reason it is important to be able to determine the structure of polytopes and polyhedra, and to have a uniform notation that describes them. The two techniques are explained and demonstrated on the examples of polytopes generated by the finite Coxeter groups A3 and H3.

Квантовые корреляции коллоидных частиц как первопричина топологических особенностей микрокосма оксигидратных систем и их свойств

A fundamentally new method for calculating the crystallographic structures of oxyhydrate gels that transform over time is proposed. Simple features of caustics and wave fronts form two infinite series and and three exceptional features (Symmetry groups are groups of regular polyhedra in three – dimensional space, and exceptional features are symmetry groups of tetrahedron, octahedron, and icosahedron.) This follows from the Coxeter-Dynkin diagrams. The principles of the program "COXETER" for the implementation of the method of calculation of oxyhydrate crystallographic structures. Gel electric moments of higher even orders are the higher degrees of Laplace operators: sixth, eighth, tenth and so on orders. In this case, a General ratio is obtained, where new amplitudes and phases are obtained by adding oscillations with different phases, recorded formulas:, which are characterized by the values of currents depending on the spatial periodic structures, that is, an experimental connection between the current and the concentration of the nanoclusters of the system is obtained . From the analysis of the experimental data it follows that the main part of the time oscillations is determined by the light elements of the clusters, the quadrupoles of which give a relatively weak current surge with a small amplitude. Along with them there are elements of the third order, the fourth, fifth and possibly sixth. The nanoclusters are formed by the rule of "magic numbers" found experimentally in the hydroxide application in complex tin, yttrium, iron and other. The rules of "magic numbers", or the theory of magic numbers is widely used to describe nano-structures quite complicated. The role of Paterns is at the heart of everything, that is, at the heart of the structure of Reality. Reality is not collected in parts from the particles of matter in the course of evolution from the past to the future, and is all at once from the past to the future for a given pattern, that is, for specific PATTERNS, as defined by quantum theory. Irregularities and partial randomization of the structure of the core grids form gel defects, to which small mobile clusters are attracted by electrostatic or electromagnetic forces, which are then adsorbed and located on the "defects" in accordance with their dipole moments. This circumstance is determined by the values of"magic numbers". Cluster magic structures have a layered structure. The interior region such fulleroids be, for example, medium-structured clusters and their multipoles and octupole. If at the same time multipoles of the same name, it inevitably leads to fluctuations of cluster flows. The areas inside the fulleroid will not be filled completely; they always remain empty. This will also lead to permanent oscillations of the cluster “filling” and is related to the filling of the fluctuations of the cluster environment. Knowing the padding settings fulleroid (fluctuations) can be identified, what are its physico-chemical characteristics. Linear ordered REALITY, or in other terminology, historical oxyhydrate sequence – is the result of experimental wave interference (table 4) sums of historical (time) epochs-waves of equations Moreover, the Function satisfies the condition of the Wheeler-DeWitt equation, which allows to describe the birth of a 4-dimensional space-time gel oxyhydrate phase.

Binary Icosahedral Group and 600-Cell

In this article, we have an explicit description of the binary isosahedral group as a 600-cell. We introduce a method to construct binary polyhedral groups as a subset of quaternions H via spin map of SO(3). In addition, we show that the binary icosahedral group in H is the set of vertices of a 600-cell by applying the Coxeter–Dynkin diagram of H4.

Algebraic curves uniformized by congruence subgroups of triangle groups

We construct certain subgroups of hyperbolic triangle groups which we call \congruence subgroups. These groups include the classical congruence subgroups of SL2(Z), Hecke triangle groups, and 19 families of arithmetic triangle groups associated to Shimura curves. We determine the eld of moduli of the curves associated to these groups and thereby realize the groups PSL2(Fq) and PGL2(Fq) regularly as Galois groups.

SU(5) grand unified theory, its polytopes and 5-fold symmetric aperiodic tiling

We associate the lepton–quark families with the vertices of the 4D polytopes 5-cell [Formula: see text] and the rectified 5-cell [Formula: see text] derived from the [Formula: see text] Coxeter–Dynkin diagram. The off-diagonal gauge bosons are associated with the root polytope [Formula: see text] whose facets are tetrahedra and the triangular prisms. The edge-vertex relations are interpreted as the [Formula: see text] charge conservation. The Dynkin diagram symmetry of the [Formula: see text] diagram can be interpreted as a kind of particle-antiparticle symmetry. The Voronoi cell of the root lattice consists of the union of the polytopes [Formula: see text] whose facets are 20 rhombohedra. We construct the Delone (Delaunay) cells of the root lattice as the alternating 5-cell and the rectified 5-cell, a kind of dual to the Voronoi cell. The vertices of the Delone cells closest to the origin consist of the root vectors representing the gauge bosons. The faces of the rhombohedra project onto the Coxeter plane as thick and thin rhombs leading to Penrose-like tiling of the plane which can be used for the description of the 5-fold symmetric quasicrystallography. The model can be extended to [Formula: see text] and even to [Formula: see text] by noting the Coxeter–Dynkin diagram embedding [Formula: see text]. Another embedding can be made through the relation [Formula: see text] for more popular [Formula: see text]. Appendix A includes the quaternionic representations of the Coxeter–Weyl groups [Formula: see text] which can be obtained directly from [Formula: see text] by projection. This leads to relations of the [Formula: see text] polytopes with the quasicrystallography in 4D and [Formula: see text] polytopes. Appendix B discusses the branching of the polytopes in terms of the irreducible representations of the Coxeter–Weyl group [Formula: see text].

A note on distinguished bases of singularities

For an isolated hypersurface singularity which is neither simple nor simple elliptic, it is shown that there exists a distinguished basis of vanishing cycles which contains two basis elements with an arbitrary intersection number. This implies that the set of Coxeter–Dynkin diagrams of such a singularity is infinite, whereas it is finite for the simple and simple elliptic singularities. For the simple elliptic singularities, it is shown that the set of distinguished bases of vanishing cycles is also infinite. We also show that some of the hyperbolic unimodal singularities have Coxeter–Dynkin diagrams like the exceptional unimodal singularities.

Symmetry of the Pyritohedron and Lattices

The pyritohedron consisting of twelve identical but non regular pentagonal faces and its dual pseudoicosahedron that possess the pyritohedral (Th) symmetry play an essential role in understanding the crystallographic structures with the pyritohedral symmetry. The pyritohedral symmetry takes a simpler form in terms of quaternionic representation. We discuss the 3D crystals with the pyritohedral symmetry which can be derived from the Coxeter-Dynkin diagram of D3.

On the arithmetic dimension of triangle groups

Let $\Delta=\Delta(a,b,c)$ be a hyperbolic triangle group, a Fuchsian group obtained from reflections in the sides of a triangle with angles $\pi/a,\pi/b,\pi/c$ drawn on the hyperbolic plane. We define the arithmetic dimension of $\Delta$ to be the number of split real places of the quaternion algebra generated by $\Delta$ over its (totally real) invariant trace field. Takeuchi has determined explicitly all triples $(a,b,c)$ with arithmetic dimension $1$, corresponding to the arithmetic triangle groups. We show more generally that the number of triples with fixed arithmetic dimension is finite, and we present an efficient algorithm to completely enumerate the list of triples of bounded arithmetic dimension.

Faces of root polytopes in all dimensions.

In this paper the root polytopes of all finite reflection groups W with a connected Coxeter-Dynkin diagram in {\bb R}^n are identified, their faces of dimensions 0 ≤ d ≤ n - 1 are counted, and the construction of representatives of the appropriate W-conjugacy class is described. The method consists of recursive decoration of the appropriate Coxeter-Dynkin diagram [Champagne et al. (1995). Can. J. Phys. 73, 566-584]. Each recursion step provides the essentials of faces of a specific dimension and specific symmetry. The results can be applied to crystals of any dimension and any symmetry.

To the Hilbert class field from the hypergeometric modular function

In this article we make an explicit approach to the problem: “For a given CM field M, construct its maximal unramified abelian extension C(M) by the adjunction of special values of certain modular functions” in some restricted cases with [M:Q]≥4. We make our argument based on Shimura's main result on the complex multiplication theory of his article in 1967. His main result treats CM fields embedded in a quaternion algebra B over a totally real number field F. We determine the modular function which gives the canonical model for all B's coming from arithmetic triangle groups. That is our main theorem. As its application, we make an explicit case-study for B corresponding to the arithmetic triangle group Δ(3,3,5). By using the modular function of K. Koike obtained in 2003, we show several examples of the Hilbert class fields as an application of our theorem to this triangle group.

On conjugacy of Cartan subalgebras in extended affine Lie algebras

That finite-dimensional simple Lie algebras over the complex numbers can be classified by means of purely combinatorial and geometric objects such as Coxeter–Dynkin diagrams and indecomposable irreducible root systems, is arguably one of the most elegant results in mathematics. The definition of the root system is done by fixing a Cartan subalgebra of the given Lie algebra. The remarkable fact is that (up to isomorphism) this construction is independent of the choice of the Cartan subalgebra. The modern way of establishing this fact is by showing that all Cartan subalgebras are conjugate.For symmetrizable Kac–Moody Lie algebras, with the appropriate definition of Cartan subalgebra, conjugacy has been established by Peterson and Kac. An immediate consequence of this result is that the root systems and generalized Cartan matrices are invariants of the Kac–Moody Lie algebras. The purpose of this paper is to establish conjugacy of Cartan subalgebras for extended affine Lie algebras; a natural class of Lie algebras that generalizes the finite-dimensional simple Lie algebra and affine Kac–Moody Lie algebras.

On the lattice-geometry and birational group of the six-point multi-ratio equation

The inherent self-consistency properties of the six-point multi-ratio equation allow it to be considered on a domain associated with a T-shaped Coxeter–Dynkin diagram. This extends the Kadomtsev–Petviashvili lattice, which has A N symmetry, and incorporates also Korteweg–de Vries-type dynamics on a sub-domain with D N symmetry, and Painlevé dynamics on a sub-domain with E ~ 8 symmetry. More generally, it can be seen as a distinguished representation of Coble’s Cremona group associated with invariants of point sets in projective space.

Imaginary Verma modules and Kashiwara algebras for [formula omitted

We consider imaginary Verma modules for quantum affine algebra Uq(gˆ), where gˆ has Coxeter–Dynkin diagram of ADE type, and construct Kashiwara type operators and the Kashiwara algebra Kq. We show that a certain quotient Nq− of Uq(gˆ) is a simple Kq-module.

A note on bimodal singularities and mirror symmetry

We discuss the relation between transposition mirror symmetry of Berglund and Hübsch for bimodal singularities and polar duality of Batyrev for associated toric K3 hypersurfaces. We also show that homological mirror symmetry for singularities implies the geometric construction of Coxeter–Dynkin diagrams of bimodal singularities by Ebeling and Ploog.

The structure of the full icosahedral group

We consider the full icosahedral group, which is the Coxeter group of type H_3. The Coxeter groups appear naturally in geometry and algebra. In 1935, the finite Coxeter groups were classified by Coxeter in terms of Coxeter-Dynkin diagrams. We remark that the affine extensions of the Coxeter groups of types H are related to quasicrystals with tenfold symmetry. Our approach to understanding the structure of Coxeter groups is the noncommutative Groebner basis theory, which is called the Groebner-Shirshov basis theory. By completing the relations coming from a presentation of the Coxeter group, we find a Groebner-Shirshov basis to obtain a set of standard monomials. Especially, for the Coxeter group of type H_3, its Groebner-Shirshov basis and the corresponding standard monomials are constructed. Thus, we understand the algebra structure of the group algebra C[H_3], which is not commutative.

Faces of Platonic solids in all dimensions.

This paper considers Platonic solids/polytopes in the real Euclidean space R(n) of dimension 3 ≤ n < ∞. The Platonic solids/polytopes are described together with their faces of dimensions 0 ≤ d ≤ n - 1. Dual pairs of Platonic polytopes are considered in parallel. The underlying finite Coxeter groups are those of simple Lie algebras of types A(n), B(n), C(n), F4, also called the Weyl groups or, equivalently, crystallographic Coxeter groups, and of non-crystallographic Coxeter groups H3, H4. The method consists of recursively decorating the appropriate Coxeter-Dynkin diagram. Each recursion step provides the essential information about faces of a specific dimension. If, at each recursion step, all of the faces are in the same Coxeter group orbit, i.e. are identical, the solid is called Platonic. The main result of the paper is found in Theorem 2.1 and Propositions 3.1 and 3.2.