Related papers: Structures of Coincidence Symmetry Groups
The concepts of symmetry and symmetry groups are at the heart of several developments in modern theoretical and mathematical physics. The present paper is devoted to a number of selected topics within this framework: Euclidean and rotation…
Every isometry of a finite dimensional euclidean space is a product of reflections and the minimum length of a reflection factorization defines a metric on its full isometry group. In this article we identify the structure of intervals in…
Symmetries impose structure on the Hilbert space of a quantum mechanical model. The mathematical units of this structure are the irreducible representations of symmetry groups and I consider how they function as conceptual units of…
We present a method of constructing discrete integrable systems with crystallographic reflection group (Weyl) symmetries, thus clarifying the relationship between different discrete integrable systems in terms of their symmetry groups.…
We study quasi-isometric embeddings of symmetric spaces and non-uniform irreducible lattices in semisimple higher rank Lie groups. We show that any quasi-isometric embedding between symmetric spaces of the same rank can be decomposed into a…
Let $E$ be a two-dimensional \'etale algebra over a non-Archimedean local field $K$ of characteristic zero. We show that the unitary group of a non-degenerate hermitian lattice over $E$ is generated by symmetries and rescaled Eichler…
Group lattices (Cayley digraphs) of a discrete group are in natural correspondence with differential calculi on the group. On such a differential calculus geometric structures can be introduced following general recipes of noncommutative…
Based on a fundamental symmetry between space, time, mass and charge, a series of group structures of physical interest is generated, ranging from C2 to E8. The most significant result of this analysis is a version of the Dirac equation…
In the study of pattern formation in symmetric physical systems a 3-dimensional structure in thin domains is often modelled as 2-dimensional one. We are concerned with functions in $R^3$ that are invariant under the action of a…
The coincidence problem for three-dimensional discrete structures with icosahedral symmetry is reinvestigated. We present a parametric description of the coincidence rotations based on special quaternions, called icosian numbers. In…
The problem of computing the index of a coincidence isometry of the hyper cubic lattice $\mathbb{Z}^{n}$ is considered. The normal form of a rational orthogonal matrix is analyzed in detail, and explicit formulas for the index of certain…
We present some of the group theoretic properties of reversing symmetry groups, and classify their structure in simple cases that occur frequently in several well-known groups of dynamical systems.
We announce various results concerning the structure of compactly generated simple locally compact groups. We introduce a local invariant, called the structure lattice, which consists of commensurability classes of compact subgroups with…
Diagonal groups are one of the classes of finite primitive permutation groups occurring in the conclusion of the O'Nan-Scott theorem. Several of the other classes have been described as the automorphism groups of geometric or combinatorial…
The degree of symmetry of a combinatorial object, such as a lattice path, is a measure of how symmetric the object is. It typically ranges from zero, if the object is completely asymmetric, to its size, if it is completely symmetric. We…
We start by introducing the basics of configurations of points and lines, and then move into discussing symmetry groups of these configurations. Specifically, we explore how we might classify the symmetries of $(9_3)$ and $(10_3)$ geometric…
For real hyperbolic spaces, the dynamics of individual isometries and the geometry of the limit set of nonelementary discrete isometry groups have been studied in great detail. Most of the results were generalised to discrete isometry…
Each quiver appearing in a seed of a cluster algebra determines a corresponding group, which we call a cluster group, which is defined via a presentation. Grant and Marsh showed that, for quivers appearing in seeds of cluster algebras of…
The coincidence site lattice (CSL) problem and its generalization to Z-modules in Euclidean 3-space is revisited, and various results and conjectures are proved in a unified way, by using maximal orders in quaternion algebras of class…
Four-dimensional twisted group lattices are used as models for space-time structure. Compared to other attempts at space-time deformation, they have two main advantages: They have a physical interpretation and there is no difficulty in…