Wreath Products Of Groups Research Articles

Computer generation of character tables of generalized wreath product groups

AbstractAlgorithms and computer codes in FORTRAN 77 are developed for the computation of characters of generalized wreath product groups through generating function methods. Characters of generalized wreath product groups of symmetric groups are computed in terms of the characters of much smaller composing groups using plethysms of S‐functions. The developed codes were tested for many generalized wreath product groups. The character tables of generalized wreath products, S2[S3, S2, S2] (NMR group of hexane) and S4[S3] containing 576 and 31,104 elements, respectively, were constructed as examples.

Generators of the character tables of generalized wreath product groups

A method based on the cycle-type matrix algebra is devised to generate the character tables of generalized wreath products which are useful in describing the symmetry groups of non-rigid molecules, NMR groups, groups of non-rigid van der Waals complexes, the groups of space types in configuration interaction calculations, etc. This newly developed method is illustrated with examples.

Evolution on sequence space and tensor products of representation spaces

A connection between evolutionary mutation-selection dynamics on sequence space and linear representations of the hyper-octahedral group (interpreted as the isometry group of the space of {0, 1-sequences relative to the Hamming distance metric) and other such wreath product groups is established and studied.

Applications of caterpillar trees in chemistry and physics

The relations of caterpillar trees (which are also known as Gutman trees and benzenoid trees) to other mathematical objects such as polyhex graphs, Clar graphs, king polyominos, rook boards and Young diagrams are discussed. Potential uses of such trees in data reduction, computational graph theory, and in the ordering of graphs are considered. Combinatorial and physical properties of benzenoid hydrocarbons can be studied via related caterpillars. It thus becomes possible to study the properties of large graphs such as benzenoid (i.e. polyhex) graphs in terms of much smaller tree graphs. Generation of the cyclic structures of wreath and generalized wreath product groups through the use of caterpillar trees is illustrated.

Chemical applications of topology and group theory. XXI: Chirality in transitive skeletons and qualitative completeness [1

This paper unifies the following ideas for the study of chirality polynomials in transitive skeletons: (1) Generalization of chirality to permutation groups not corresponding to three-dimensional symmetry point groups leading to the concepts of signed permutation groups and their signed subgroups; (2) Determination of the total dimension of the chiral ligand partitions through the Frobenius reciprocity theorem; (3) Determination of signed permutation groups, not necessarily corresponding to three-dimensional point groups, of which a given ligand partition is a maximum symmetry chiral ligand partition by the Ruch-Schonhofer partial ordering, thereby allowing the determination of corresponding chirality polynomials depending only upon differences between ligand parameters; such permutation groups having the point group as a signed subgroup relate to qualitative completeness. In the case of transitive permutation groups on four sites, the tetrahedron and polarized square each have only one chiral ligand partition, but the allene and polarized rectangle skeletons each have two chiral ligand partitions related to their being signed subgroups of the tetrahedron and polarized square, respectively. The single transitive permutation group on five sites, the polarized pentagon, has a degenerate chiral ligand partition related to its being a signed subgroup of a metacyclic group with 20 elements. The octahedron has two chiral ligand partitions, both of degree six; a qualitatively complete chirality polynomial is therefore homogeneous of degree six. The cyclopropane (or trigonal prism or trigonal antiprism) skeleton is a signed subgroup of both the octahedron and a twist group of order 36; two of its six chiral ligand partitions come from the octahedron and two more from the twist group. The polarized hexagon is a signed subgroup of the same twist group but not of the octahedron and thus has a different set of six chiral ligand partitions than the cyclopropane skeleton. Two of its six chiral ligand partitions come from the above twist group of order 36 and two more from a signed permutation group of order 48 derived from the P3[P 2] wreath product group with a different assignment of positive and negative operations than the octahedron.

A note on generalised wreath product groups

AbstractGeneralised wreath products of permutation groups were discussed in a paper by Bailey and us. This note determines the orbits of the action of a generalised wreath product group on m–tuples (m ≥ 2) of elements of the product of the base sets on the assumption that the action on each component is m–transitive. Certain related results are also provided.

The characters of the wreath product group acting on the homology groups of the Dowling lattices

Let G be a finite group, n a positive integer, Q n ( G) the Dowling lattice of rank n based on G and W n the wreath product group G wr S n . It is easily seen that W n acts as a group of automorphisms of Q n ( G). This action lifts to a representation of W n on each homology group of Q n ( G). The character values of these representations are computed. Let σ be an element of W n . Consider σ as an n × n permutation matrix σ whose nonzero entries have been replaced by elements of G. If C is a cycle of σ, the weight of C is the product of the elements of G which lie in the cycle C. The type of C is the conjugacy class of G containing the weight of C. Let c l, u denote the number of l-cycles of σ of type u. The conjugacy class of σ in W n depends only on the numbers c l, u . For each i = 0, 1,…, n − 1 let ( Q n ( G)) i be the geometric lattice obtained from Q n ( G) by deleting ranks i + 1 through n − 1 (so ( Q n ( G)) n − 1 = Q n ( G)). Let β i denote the character of the representation of W n on the unique non-vanishing reduced homology group of Q n ( G)) i . For each σ ϵ W n , let B σ ( λ) be the polynomial B σ ( λ) = ∑ i = 0 n − 1 β i ( σ)) λ n − 1 − i . It is shown that B 0(λ)=(−1) n ∏ l,u C l,u!(l|C G(u)|) cl,u λ+1 F(l,u, l) − C l,u F(l,u,−λ) where F(l, u, λ) = ( −1 l¦G¦ ) ∑ t¦lb(t,u) μ(t) λ l t , and where b(t, u) is the number of solutions h ϵ G to h t = u. The formula (∗) allows for the explicit computation of the characters β i . Using this information, several facts about the characters are deduced. For example, it is shown that the trivial character appears exactly once in each β i and it is shown that β n − 1 can be realized in a simple way as a sum of induced characters.

The second centre of a wreath product of groups

The second centre of a wreath product of groups

Symmetry operators of generalized wreath products and their applications to chemical physics

AbstractSymmetry operators of generalized wreath product groups are formulated. Several applications of these operators to nonrigid molecular problems in chemical physics are outlined.

Symmetry simplifications of space types in configuration interaction induced by orbital degeneracy

AbstractSymmetry simplifications are introduced in configuration interaction (CI) by reducing the number of symmetry‐allowed space types if there is degeneracy in some of the molecular orbitals by constructing the unique space types. A new symmetry group which we call the configuration symmetry group is defined and is shown to be expressible as a generalized wreath product group. Generating functions are derived for enumerating the equivalence classes of space types. A double coset method is expounded which constructs the representatives of all equivalence classes of space types using the cycle index of generalized wreath product and the double cosets of label subgroup with generalized wreath product in the symmetric group Sn, if n is twice the number of occupied and virtual orbitals. Method is illustrated with CI using the localized orbitals of polyenes, CI in benzene, and atomic CI for several reference states.

Graph theoretical characterization of NMR groups, nonrigid nuclear spin species and the construction of symmetry adapted NMR spin functions

NMR group defined by Woodman is shown to be the generalized wreath product group of the NMR graph which is constructed with protons as vertices and edges representing nuclear couplings. The NMR graph can be expressed as a generalized composition of its quotient graph and various types. It is shown that the NMR group is the automorphism group of the NMR graph that preserves the coupling matrix defined here. Using the representation theory of generalized wreath product groups developed in an earlier paper of the present author, the character tables of NMR groups can be obtained. The character tables of certain NMR groups are presented. The quotient graph of the NMR graph is shown to be the composite particle graph. The automorphism group of the quotient NMR graph is the symmetry group of the composite particles, preserving the NMR coupling constants. An iterative algorithm is formulated for obtaining the automorphism group of the composite particle tree. A combinatorial approach is expounded for identifying the nuclear spin species of nonrigid and rigid molecules. The coalescence diagrams describing the effect of nonrigidity on nuclear spin species are introduced and obtained to exemplify the utility of the proposed method. It is shown that using the automorphism group of the quotient graph, the symmetry adapted composite particle spin functions can be constructed. We illustrate this method with 2,3-dimethylbutane. The nuclear spin Hamiltonian matrix of 2,3-dimethylbutane of order 214×214 is reduced to a matrix of order 64×64 by the composite particle method. This matrix can be blocked in the symmetry adapted basis set constructed through the automorphism group of the quotient graph into 40 matrices of order 1×1 and 12 matrices of order 2×2.

The symmetry groups of nonrigid molecules as generalized wreath products and their representations

The symmetry groups of nonrigid molecules first defined by Longuet–Higgins, in the most general cases are generalized wreath product groups. We first outline the representation theory of wreath product groups. Then the representation theory of generalized wreath product groups is developed. Several illustrative examples of NMR groups are offered. The character tables of several nonrigid molecular groups are presented. An example of a nonrigid triphenyl molecule is given to illustrate the use of the generalized wreath product in deriving optical selection rules.

A generalized wreath product method for the enumeration of stereo and position isomers of polysubstituted organic compounds

Ageneralized wreath product group is developed in the root-to-root product formalism for the enumeration of stereo and position isomers of polysubstituted organic compounds. The methods expounded here are used for enumerating the NMR signals of polysubstituted organic compounds.

Minimal Relations for Certain Wreath Products of Groups

Let p be a rational prime, G a non-trivial finite p group, and K the field of p elements, regarded as a trivial G-module according to context; then we define:d(G) = dimKH1(G, K), the minimal number of generators of G,r(G) = dimKH2(G, K),r′(G) = the minimal number of relations required to define G,where, in the last equation, it is sufficient to take the minimum over those presentations of G with d(G) generators. It is well known (see § 2) that the following inequalities hold:We shall consider only finite p-groups, so that the class of groups with r = d coincides with that consisting of those groups whose Schur multiplicator is trivial.

Topological wreath products

The object of this note is to show that under suitable restrictions some results on the wreath product of groups can be carried over to topological groups. We prove in particular the following analogue of the well-known theorem of Krasner and Kaloujnine (see for example [2] Theorem 3.5): Theorem. Let A and B be two locally compact topological groups, and let (C, ε) be an extension of A by B. If there exists a continuous left inverseof ε, that is to say a continuous mapping τ: B → C such that re is the identity on B, then there exists a continuous monomorphism of C into the topological standard wreath product of A by B.