Coincidences and colorings of lattices and Z-modules

Loquias M. Coincidences and colorings of lattices and Z-modules. Bielefeld (Germany): Bielefeld University; 2010.This thesis delves into the coincidence problem for sublattices and submodules, colorings of lattices and Z-modules, shifted lattices and shifted Z-modules, and multilattices. Moreover, the idea of a coincidence isometry of a lattice or Z-module is extended to include general (affine) isometries. The first chapter gives all the essential definitions, notations, and known results. It starts with a description of the coincidence problem for lattices and Z-modules in R^d. A discussion of the results on the coincidences of the square lattice, certain n-planar modules, cubic lattices, and hypercubic lattices follows the general treatment. Notions on colorings of lattices and Z-modules end the chapter. Even though a lattice and its sublattices have the same set of coincidence isometries, the coincidence indices and corresponding multiplicities with respect to the lattice and its sublattices are in general different. A method of computing the coincidence index of a coincidence isometry of a lattice with respect to a sublattice is formulated via properties of the coloring of the lattice determined by the sublattice. This result motivates a generalization of the idea of color symmetry to that of a color coincidence. Examples and other general results are provided to illustrate these ideas. These concepts and results comprise Chapter 2 of the thesis. In Chapter 3.1, intersections of two lattices and Z-modules that are related by any isometry are referred to as affine coincidence site lattices (ACSLs) and modules (ACSMs), respectively, and the isometries that generate them as affine coincidence isometries. The affine coincidence isometries of a lattice and Z-module, and the resulting intersections, are identified. The rest of Chapter 3 covers a related and special case: the coincidence problem for shifted lattices and shifted Z-modules. That is, after translating the lattice or Z-module G by some vector x, and upon application of a linear isometry R to the shifted lattice or shifted Z-module x+G (with respect to the origin), its intersection with x+G is considered. The (linear) coincidence isometries of x+G and the CSLs/CSMs of the shifted lattice/Z-module are determined. An extensive analysis of the coincidences of a shifted square lattice follows and is achieved by identifying the lattice with the ring of Gaussian integers. The coincidence problem is completely solved when the shift consists of an irrational component. For the remaining case, that is, when the shift may be written as a quotient of two Gaussian integers that are relatively prime, one computes for the set of coincidence rotations of the shifted square lattice using some divisibility condition involving the denominator of the shift. In both instances, the set of coincidence rotations of a shifted square lattice forms a group. An example is given where the set of coincidence isometries of a shifted square lattice is not a group. Corresponding results and an example for planar modules conclude the chapter. The final chapter of this thesis is concerned with coincidences of sets of points formed by the union of a lattice with a finite number of shifted copies of the lattice. Such sets are referred to as multilattices. The chapter starts with an analysis of the coincidences of the simplest multilattice, that is, of the union of a lattice and a shifted lattice. This leads to the solution of the coincidence problem for the diamond packing. The main result of the chapter gives the solution of the coincidence problem for general multilattices. The main problem in Chapter 2 is then revisited, where the reverse condition is now considered. More accurately, if the coincidence problem for a sublattice of a given lattice has already been solved, then what can be deduced about the CSLs and corresponding coincidence indices of the original lattice? This question is resolved by regarding the lattice as a multilattice formed by the union of the sublattice with the cosets of the sublattice. This perspective establishes a connection among the relationship between the coincidence indices of a lattice and a sublattice, color coincidences of the coloring of the lattice determined by the sublattice, and coincidences of shifted lattices. The chapter ends with a full description of the case when a sublattice is of prime index in a lattice, and the solution of the coincidence problem for certain primitive and centered rectangular lattices