Coset Representation Research Articles

Special Relativity and Its Newtonian Limit from a Group Theoretical Perspective

In this pedagogical article, we explore a powerful language for describing the notion of spacetime and particle dynamics intrinsic to a given fundamental physical theory, focusing on special relativity and its Newtonian limit. The starting point of the formulation is the representations of the relativity symmetries. Moreover, that seriously furnishes—via the notion of symmetry contractions—a natural way in which one can understand how the Newtonian theory arises as an approximation to Einstein’s theory. We begin with the Poincaré symmetry underlying special relativity and the nature of Minkowski spacetime as a coset representation space of the algebra and the group. Then, we proceed to the parallel for the phase space of a spin zero particle, in relation to which we present the full scheme for its dynamics under the Hamiltonian formulation, illustrating that as essentially the symmetry feature of the phase space geometry. Lastly, the reduction of all that to the Newtonian theory as an approximation with its space-time, phase space, and dynamics under the appropriate relativity symmetry contraction is presented. While all notions involved are well established, the systematic presentation of that story as one coherent picture fills a gap in the literature on the subject matter.

Complete spectrum of the infinite-U Hubbard ring using group theory.

We present a full analytical solution of the multiconfigurational strongly correlated mixed-valence problem corresponding to the N-Hubbard ring filled with N-1 electrons, and infinite on-site repulsion. While the eigenvalues and the eigenstates of the model are known already, analytical determination of their degeneracy is presented here for the first time. The full solution, including degeneracy count, is achieved for each spin configuration by mapping the Hubbard model into a set of Hückel-annulene problems for rings of variable size. The number and size of these effective Hückel annulenes, both crucial to obtain Hubbard states and their degeneracy, are determined by solving a well-known combinatorial enumeration problem, the necklace problem for N-1 beads and two colors, within each subgroup of the CN-1 permutation group. Symmetry-adapted solution of the necklace enumeration problem is finally achieved by means of the subduction of coset representation technique [S. Fujita, Theor. Chim. Acta 76, 247 (1989)], which provides a general and elegant strategy to solve the one-hole infinite-U Hubbard problem, including degeneracy count, for any ring size. The proposed group theoretical strategy to solve the infinite-U Hubbard problem for N-1 electrons is easily generalized to the case of arbitrary electron count L, by analyzing the permutation group CL and all its subgroups.

Symmetry-itemized enumeration of RS-stereoisomers of allenes. I. The fixed-point matrix method of the USCI approach combined with the stereoisogram approach

After the RS-stereoisomeric group \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}\) of order 16 has been defined by starting point group \(\mathbf{D}_{2d}\) of order 8, the isomorphism between \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}\) and the point group \(\mathbf{D}_{4h}\) of order 16 is thoroughly discussed. The non-redundant set of subgroups (SSG) of \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}\) is obtained by referring to the non-redundant set of subgroups of \(\mathbf{D}_{4h}\). The coset representation for characterizing the orbit of the four positions of an allene skeleton is clarified to be \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}(/\mathbf{C}_{s\widetilde{\sigma }\widehat{I}})\), which is closely related to the \(\mathbf{D}_{4h}(/\mathbf{C}_{2v}^{\prime \prime \prime })\). According to the unit-subduced-cycle-index (USCI) approach (Fujita, Symmetry and combinatorial enumeration of chemistry. Springer, Berlin 1991), the subduction of \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}(/\mathbf{C}_{s\widetilde{\sigma }\widehat{I}})\) is examined so as to generate unit subduced cycle indices with chirality fittingness (USCI-CFs). Then, the fixed-point matrix method of the USCI approach is applied to the USCI-CFs. Thereby, the numbers of quadruplets are calculated in an itemized fashion with respect to the subgroups of \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}\). After the subgroups of \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}\) are categorized into types I–V, type-itemized enumeration of quadruplets is conducted to illustrate the versatility of the stereoisogram approach.

Symmetry-itemized enumeration of quadruplets of RS-stereoisomers: I—the fixed-point matrix method of the USCI approach combined with the stereoisogram approach

The RS-stereoisomeric group $$\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$$ is examined to characterize quadruplets of RS-stereoisomers based on a tetrahedral skeleton and found to be isomorphic to the point group $$\mathbf{O}_{h}$$ of order 48. The non-redundant set of subgroups (SSG) of $$\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$$ is obtained by referring to the non-redundant SSG of $$\mathbf{O}_{h}$$ . The coset representation for characterizing the orbit of the four positions of the tetrahedral skeleton is clarified to be $$\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}(/\mathbf{C}_{3v\widetilde{\sigma }\widehat{I}})$$ , which is closely related to the $$\mathbf{O}_{h}(/\mathbf{D}_{3d})$$ . According to the unit-subduced-cycle-index (USCI) approach (Fujita in Symmetry and combinatorial enumeration in chemistry. Springer, Berlin, 1991), the subdution of $$\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}(/\mathbf{C}_{3v\widetilde{\sigma }\widehat{I}})$$ is examined so as to generate unit subduced cycle indices with chirality fittingness (USCI-CFs). The fixed-point matrix method of the USCI approach is applied to the USCI-CFs. Thereby, the numbers of quadruplets are calculated in an itemized fashion with respect to the subgroups of $$\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$$ . After the subgroups of $$\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$$ are categorized into types I–V, type-itemized enumeration of quadruplets is conducted to illustrate the versatility of the stereoisogram approach.

Conditionally Cycle-Free Graphical Models for Coset Codes

Conditionally cycle-free graphical models (i.e., cyclic graphical models which become cycle-free after conditioning on a subset of the hidden variables) are constructed for coset codes. Following the description of a general construction procedure, examples of a number of families of codes-including first-order Reed-Muller (RM) and the Delsarte-Goethals codes - are provided for which the proposed procedure yields optimal soft-in soft-out (SISO) decoding algorithms that are less complex than the best known trellis-based algorithms. In the case of the first-order RM codes, which have a recursive coset construction, the optimal SISO decoding algorithm that results when the proposed construction is applied repeatedly is denoted recursive coset representation (RCR) decoding. Connections are made between RCR decoding and existing algorithms that exploit fast Hadamard transforms. Finally, the utility of the proposed decoding algorithms are supported by a practically motivated application: the construction of serially concatenated codes that have high rates and low error floors. Extended Hamming codes are proposed as outer codes in such constructions with efficient decoding employing the SISO algorithms developed herein.

Statistics of isomorphism types in free products

Let Γ be a free product of finitely many finite- and infinite-cyclic groups. For a subgroup Δ of finite index given by its coset representation we compute its isomorphism type, i.e., its decomposition as a free product of finite- and infinite-cyclic groups. We determine the set of isomorphism types realized by finite-index subgroups, the asymptotics of the subgroup numbers with prescribed isomorphism types, and the distribution of the isomorphism types among subgroups of fixed index. Apart from group-theoretic arguments, the proofs of the present paper make use of asymptotic, combinatorial, and probabilistic ideas and techniques.

Stereogenicity/Astereogenicity as Global/Local Permutation-Group Symmetry and Relevant Concepts for Restructuring Stereochemistry

Molecules of ligancy 4 that have been derived from an allene, an ethylene, a tetrahedral, and a square-planar skeleton have been investigated to show that their symmetries are dually and distinctly controlled by point groups and permutation groups. Insomuch as the point-group symmetry was exhibited to control the chirality/achirality of a molecule, sphericity in a molecule, and enantiomeric relationship between molecules [S. Fujita, J. Am. Chem. Soc. 112 (1990) 3390], the permutation-group symmetry has been now clarified to control the stereogenicity of a molecule, tropicity in a molecule, and diastereomeric relationship between molecules. To characterize permutation groups, proper and improper permutations have been defined by comparing proper and improper rotations. Thereby, such permutation groups are classified into stereogenic and astereogenic ones. After a coset representation (CR) of a permutation group has been ascribed to an orbit (equivalence class), the tropicity of the orbit has been defined in term of the global stereogenicity and the local stereogenicity of the CR. As a result, the conventional “stereogenicity” has now been replaced by the concept local stereogenicity of the present investigation. The terms homotropic, enantiotropic, and hemitropic are coined and used to characterize prostereogenicity. Thus, a molecule is defined as being prostereogenic if it has at least one enantiotropic orbit. Since this definition has been found to be parallel with the definition of prochirality, relevant concepts have been discussed with respect to the parallelism between stereogenicity and chirality in order to restructure the theoretical foundation of stereochemistry and stereoisomerism. The derivation of the skeletons has been characterized by desymmetrization due to the subduction of CRs. The Cahn–Ingold–Prelog (CIP) system has been discussed from the permutational point of view to show that it specifies diastereomeric relationships only. The apparent specification of enantiomeric relationships by the CIP system has been shown to stem from the fact that diastereomeric relationships and enantiomeric ones overlap occasionally in case of tetrahedral molecules.

Graphical Models of Characters of Groups

A graphical method of generating one- and (some) two-dimensional characters (Γ) has been developed on the basis of a reduced homomer set, which has been derived from a new concept of negative graphs. Thus, a homomer set H[G(/G i )]={h1,...,hd−1,h d } (d=|G|/|G i |) has been generated from a regular body of G so that it has been governed by the coset representation G(/G i ). The homomer set has been reduced into a reduced homomer set ℋ′[Γ]={h1,...,hd−1}, where we have placed h d ≡−(h1+⋅⋅⋅+hd−1) in terms of negative graphs. The action of the symmetry operations of G on the reduced homomer set ℋ′[Γ] has graphically generated a one- or (some) two-dimensional character (Γ). The versatility of the graphical method has been tested by using C3v, D2h, C2h, C2v, D3h, and C3h as examples. The graphical method has been compared with an alternative algebraic generation using marks (or markaracters), i.e., Γ=G(/G i )−G(/G).

Sphericity beyond Topicity in Characterizing Stereochemical Phenomena. Novel Concepts Based on Coset Representations and Their Subductions

AbstractFor Abstract see ChemInform Abstract in Full Text.

Sphericity beyond Topicity in Characterizing Stereochemical Phenomena. Novel Concepts Based on Coset Representations and Their Subductions

Abstract Fundamental concepts proposed for comprehending stereochemical phenomena are described in order to bring out a more systematized format to conventional concepts and terminologies. The concept of coset representation (CR), the symbol of which is G(/Gi), is intuitively introduced by considering the global symmetry G and the local symmetry Gi for an orbit of stereochemically equivalent objects (e.g. atoms in a molecule). The SCR (set-of-coset-representations) notation for classifying molecular symmetries is proposed to remedy the usual insufficient classification based on point groups. According to the chirality/achirality of G and Gi, the sphericity concept (homospheric, enantiospheric, and hemispheric) and the relevant concept of chirality fittingness are proposed to specify the stereochemical phenomena of the G(/Gi)-orbit. The sphericity terms are shown to be superior to the well-known topicity terms (homotopic, enantiotopic, and heterotopic; chirotopic and achirotopic) especially in comprehending complicated stereochemical phenomena. In fact, the topicity terms as well as the terms “stereogenicity and prostereogenicity” and “prochirality” can be derived subsidiarily from the sphericity terms. The concept of subduction of CRs, for which the symbol G(/Gi) ↓ Gi has been coined, is proposed to characterize the desymmetrization of molecules. Thereby, the design of high-symmetry molecules is discussed in term of desymmetrization by atom replacement and by bond replacement. In order to characterize the stereochemistry of non-rigid molecules with rotatable ligands, the concept of proligand/promolecule is proposed; here biphenyl derivatives, methane derivatives, ethane derivatives, and ferrocene derivatives are examined in terms of matched and mismatched molecules. Applications to combinatorial enumeration, symmetry adapted functions, flexible six-membered cyclic compounds, and symmetry numbers are also described.

Stereochemistry and Stereoisomerism Characterized by the Sphericity Concept

Abstract The sphericity concept proposed for specifying stereochemistry in a molecule (S. Fujita, J. Am. Chem. Soc., 112, 3390 (1990)) has been extended to investigate stereoisomerism among molecules. The new matter of the present approach is to characterize the global symmetries of molecules as the “local symmetries of stereoisomerism”. Thereby, stereochemistry and stereoisomerism have been discussed on a common basis. Promolecules, which have been generated as stereochemical models of molecules by placing proligands (structureless ligands with chirality/achirality) on the vertices of a tetrahedral skeleton, have been analyzed by a permutation-group approach as well as by a point-group one. The skeleton has been considered to belong to the symmetric group of degree 4 (S[4]) as well as to the isomorphic point group Td. The chirality fittingness derived from the sphericity concept has been applied to the characterization of local symmetries of promolecule, where two types of Young’s tableaus have been compared. Thus, Young’s tableaus of symmetry have been introduced to treat the ligand packing based on the chirality fittingness. These tableaus have been compared with Young’s tableaus of permutation, which have taken no account of such chirality fittingness. The two types of Young’s tableaus have been applied to the enumeration of tetrahedral isomers under the observance and the violation of chirality fittingness. This enumeration has enabled us to clarify the quantitative aspect of the sphericity concept in characterizing isomer equivalence. Thereby, equivalent isomers under a point-group symmetry have been shown to construct an orbit of stereoisomers that is ascribed to a coset representation. Homomeric, enantiomeric, and diastereomeric relationships between stereoisomers have been discussed by means of homospheric, enantiospheric and hemispheric orbits of stereoisomers. Skeleton-based and ligand-based categories for enantiomers and diastereomers have been discussed. The stereogenicity and the prostereogenicity of the Chan–Ingold–Prelog system have been related to Young’s tableaus of permutation.

An alternative classical approach to the one-electron molecular orbital theory: the coset representation

Electronic populations and signs of the linear combination coefficients in the one-electron molecular orbital (MO) theory are interpreted classically as the analogs of action-angle variables. It is shown that in the coset representation algorithm there are action-angle variables that can substitute for these quantities. Hence, one has an alternative algorithm to the one-electron MO method, the classical coset approach. The extension to the multi-site system is formulated which lays the foundation for possible applications. The system of one electron in a linear lattice with random site energies is shown mimicking the Anderson localization.

Desymmetrization of Achiral Skeletons by Mono-Substitution. Its Characterization by Subduction of Coset Representations

Abstract A set of equivalent positions in an achiral skeleton is regarded as an orbit as-signed to a coset representation G(/Gi), where the subgroup Gi is Cnv or Cn. Desymmetrization of the achiral skeleton by substituting a single ligand is examined by virtue of the subduction of the coset representation G(/Gi) ↓ Gi, which is determined to be αGi(/Gi) + βGi(/Gk(i)). The multiplicities (α and β) of the resulting coset representations are calculated from the data of the participating groups and the normalizer of Gi.

Quasiparticle operators with non-Abelian braiding statistics

We study the gauge invariant fermions in the fermion coset representation of SU( N) k Wess-Zumino-Witten models which create, by construction, the physical excitations (quasiparticles) of the theory. We show that they provide an explicit holomorphic factorization of SU( N) k Wess-Zumino-Witten primaries and satisfy non-Abelian braiding relations.

Global topological approach to highly excited vibration: a case study of H 2O, CH 2Br 2 and CD 2Br 2

A global topological approach based on the algebraic coset representation to study highly excited molecular vibration is proposed. The algorithm allows us to elucidate ample highly excited vibrational dynamics from a global viewpoint. Topics for highly excited states studied include (1) global topological structures, (2) global symmetric and antisymmetric characters, (3) resonance overlaps and chaotic motion, (4) action transfer coefficients and (5) survival probabilities denoting the decaying of the action stored in a mode, with sample calculations for H 2O, CH 2Br 2 and CD 2Br 2.

Pseudo-point groups and chronality in enumeration of reaction pairs

The pseudo-point group $$\tilde D_{6h}$$ is constructed to characterize the symmetry of a basic pair of hexagonal reaction graphs having no par-bonds on its edges. Any pairs of reaction graphs (reaction pairs) are considered to be obtained by adding par-bonds to the edges of the basic pair; they are counted by the USCI (unit-subduced-cycle-index) approach. Thus, the six edges of the basic pair are assigned to the coset representation $$\tilde D_{6h}$$ (/C 2v ). After the subduction of the $$\tilde D_{6h}$$ is calculated, the partial-cycle-index method of the USCI approach is applied to the combinatorial enumeration of reactions pairs. Reaction pairs are classified to two categories, i.e. isoenergetic and anisoenergetic. An isoenergetic pair is concluded to be a self-reaction pair, while an anisoenergetic pair corresponds to a non-self-reaction pair. The concept of chronality is also discussed to clarify the symmetrical nature of the resulting orbits.

Subsymmetry-Itemized Enumeration of Flexible Cyclohexane Derivatives

Abstract The symmetry of a pair of two chair-forms of cyclohexane is represented by the pseudo-point group of order 24. Preparation of the mark table of the group shows the twelve substitution positions of the pair to be governed by the coset representation (/Cs). After the calculation of subduction of the (/Cs), cyclohexane derivatives are combinatorially enumerated by the USCI (unit-subduced-cycle-index) approach. A generating-function method and the elementary superposition theorem are used, giving values itemized with respect to molecular formulas and subsymmetries of . Since pseudo-point groups can be classified into iso- and anisoenergetic groups as well as into achiral and chiral groups, four categories (isoenergetic-achiral, isoenergetic-chiral (Type II), anisoenergetic-achiral (Type III), and anisoenergetic-chiral (Type IV)) are generated. The isoenergetic-achiral case is further subdivided into two cases (Type I and I′). Several pairs are illustrated and discussed in the light of this classification. The concept of chronality is also discussed.

Theoretical Foundation of Prochirality. Chirogenic Sites in an Enantiospheric Orbit

Abstract Prochirality is one of the inherent properties of an enantiospheric orbit that is designated by a coset representation G(/Gi). Desymmetrization of the G(/Gi) orbit is discussed to present a new concept “H-chirogenic site”. The chirogenic site in the G(/Gi) orbit is capable of producing a chiral H-molecule under the influence of an appropriate chiral reagent. The condition for the existence of such a chirogenic site is discussed. A convenient method using a desymmetrization lattice is presented for determining chirogenic sites.

Anisochrony in Rotatable Ligands and Symmetry Non-Equivalence Characterized by Subduction of Coset Representations

Abstract A general method of characterizing symmetry equivalence and non-equivalence in a molecule is developed on the basis of the one-one correspondence between an orbit and a coset representation. This method provides us with a theoretical foundation to understand anisochrony in rotatable ligands. A given molecule is regarded as a combination of a three-dimensional skeleton and ligands. The ligand is referred to as a segment or a fragment according to its environment. Thus, the fragment is defined as a ligand-in-isolation; and the segment denotes a ligand built into a molecule. The symmetry of the ligand as a fragment is restricted to the local symmetry. This restriction is examined by using methyl and methylene ligands as examples. A methyl fragment has a homospheric C3v(/Cs) orbit that consists of three hydrogen atoms. On the other hand, a methylene fragment has an enantiospheric Cs(/C1) orbit that contains two hydrogen atoms. Such an orbit is incorporated into various molecules to afford restricted orbits, which are characterized by subduction of coset representations. The restricted orbits are discussed in terms of global and local orbits. This discussion provides a general approach to rationalize various types of anisochronies.

Promolecules for characterizing stereochemical relationships in non-rigid molecules

Proligands are defined as hypothetical ligands which are structureless but have chirality. A promolecule consists of a skeleton and such proligands. Promolecules based on a methane and an allene skeleton are enumerated. A molecule can be constructed by a method in which proligands on such a promolecule are replaced by ligands that have three-dimensional structures. This construction is controlled by a coset representation that governs an orbit of such proligands. Thereby, the resulting molecules are classified into matched and mismatched molecules. The matched molecules retain the symmetries of the starting promolecules; on the other hand, the mismatched molecules do not. Modes of such desymmetrizations are rationalized by subduction of coset representations. The concept of prochirality is also discussed.