Coset Representations Research Articles

Higher spin currents with manifest SO(4) symmetry in the large 𝒩 = 4 holography

The large [Formula: see text] nonlinear superconformal algebra is generated by six spin-[Formula: see text] currents, four spin-[Formula: see text] currents and one spin-[Formula: see text] current. The simplest extension of these [Formula: see text] currents is described by the [Formula: see text] higher spin currents of spins [Formula: see text]. In this paper, by using the defining operator product expansions (OPEs) between the [Formula: see text] currents and [Formula: see text] higher spin currents, we determine the [Formula: see text] higher spin currents (the higher spin-[Formula: see text] currents were found previously) in terms of affine Kac–Moody spin-[Formula: see text], one currents in the Wolf space coset model completely. An antisymmetric second rank tensor, three antisymmetric almost complex structures or the structure constant are contracted with the multiple product of spin-[Formula: see text] currents. The eigenvalues are computed for coset representations containing at most four boxes, at finite [Formula: see text] and [Formula: see text]. After calculating the eigenvalues of the zeromode of the higher spin-[Formula: see text] current acting on the higher representations up to three (or four) boxes of Young tableaux in [Formula: see text] in the Wolf space coset, we obtain the corresponding three-point functions with two scalar operators at finite [Formula: see text]. Furthermore, under the large [Formula: see text] ’t Hooft-like limit, the eigenvalues associated with any boxes of Young tableaux are obtained and the corresponding three-point functions are written in terms of the ’t Hooft coupling constant in simple form in addition to the two-point functions of scalars and the number of boxes.

Twisted sectors from plane partitions

Twisted sectors arise naturally in the bosonic higher spin CFTs at their free points, as well as in the associated symmetric orbifolds. We identify the coset representations of the twisted sector states using the description of W_\infty representations in terms of plane partitions. We confirm these proposals by a microscopic null-vector analysis, and by matching the excitation spectrum of these representations with the orbifold prediction.

The so-Kazama-Suzuki models at large level

The large level limit of the N = 2 SO(2N) Kazama-Suzuki coset models is argued to be equivalent to the orbifold of 4N free fermions and bosons by the Lie group SO(2N) × SO(2). In particular, it is shown that the untwisted sector of the continuous orbifold accounts for a certain closed subsector of the coset theory. Furthermore, the ground states of the twisted sectors are identified with specific coset representations, and this identification is checked by various independent arguments.

On the Countable Measure-Preserving Relation Induced on a Homogeneous Quotient by the Action of a Discrete Group

We consider a countable discrete group $G$ acting ergodicaly and a.e. freely, by measure-preserving transformations, on an infinite measure space $(\mathcal X,\nu)$ with $\sigma$-finite measure $\nu$. Let $\Gamma \subseteq G$ be an almost normal subgroup with fundamental domain $F\subseteq \mathcal X $ of finite measure. Let $\mathcal R_G$ be the countable measurable equivalence relation on $\mathcal X$ determined by the orbits of $G$. Let $\mathcal R_G| F$ be its restriction to $F$. We find an explicit presentation, by generators and relations, for the von Neumann algebra associated, by the Feldman-Moore (\cite{FM}) construction, to the relation $\mathcal R_G|_F$. The generators of the relation $\mathcal R_G|_F$ are a set of transformations of the quotient space $F\cong \mathcal X/ \Gamma$, in a one to one correspondence with the cosets of $\Gamma$ in $G$. We prove that the composition formula for these transformations is an averaged version, with coefficients in $L^\infty(F,\nu)$, of the Hecke algebra product formula (\cite{BC}). In the situation $G = PGL_2(\mathbb Z[\frac1p])$, $\Gamma=PSL_2(\mathbb Z)$, $p\geq 3$ prime number, the relation $\mathcal R_G|_F$ is the equivalence relation associated to a free, measure-preserving action of a free group on $(p+1)/2$ generators on $F$ (\cite{Ad},\cite {Hj}). We use the coset representations of the transformations generating $\mathcal R_G|_F$ to find a canonical treeing (\cite{Ga}).

Stereoisograms for three-membered heterocycles: I. Symmetry-itemized enumeration of oxiranes under an RS-stereoisomeric group

For the purpose of characterizing the stereochemistry and stereoisomerism of oxirane derivatives, the RS-stereoisomeric group $$\mathbf{C}_{2v\widetilde{\sigma }\widehat{I}}$$ of order 8 has been defined by starting point group $$\mathbf{C}_{2v}$$ of order 4, which specifies the geometric features of an oxirane skeleton. The isomorphism between $$\mathbf{C}_{2v\widetilde{\sigma }\widehat{I}}$$ and the point group $$\mathbf{D}_{2h}$$ of order 8 is discussed algebraically and diagrammatically. The data necessary to combinatorial enumeration under $$\mathbf{C}_{2v\widetilde{\sigma }\widehat{I}}$$ , e.g., the non-redundant set of subgroups, the subduction of coset representations, and the inverse of the mark table, are prepared by referring to the data of $$\mathbf{D}_{2h}$$ . The fixed-point-matrix method and the partial-cycle-index method, which have been originally developed to accomplish combinatorial enumeration under point groups in the unit-subduced-cycle-index approach (Fujita in Symmetry and combinatorial enumeration in chemistry. Springer, Berlin, 1991), are extended and applied to the combinatorial enumeration of oxirane derivatives under the RS-stereoisomeric group $$\mathbf{C}_{2v\widetilde{\sigma }\widehat{I}}$$ . Thereby, the numbers of inequivalent quadruplets are calculated in an itemized fashion with respect to the subgroups of $$\mathbf{C}_{2v\widetilde{\sigma }\widehat{I}}$$ , where each quadruplet contained in a stereoisogram is counted once. Such quadruplets are further categorized into five types (type I–V). The enumeration of oxiranes under the point group $$\mathbf{C}_{2v}$$ as well as under the RS-permutation group $$\mathbf{C}_{2\widetilde{\sigma }}$$ is also conducted and the results are compared with those of the RS-stereoisomeric group $$\mathbf{C}_{2v\widetilde{\sigma }\widehat{I}}$$ .

Stereoisograms for reorganizing the theoretical foundations of stereochemistry and stereoisomerism. Part 3: rational avoidance of misleading standpoints for pro-R/pro-S-descriptors

The stereoisogram approach is introduced to settle the misleading terminology due to ‘prochirality’ in modern stereochemistry. After the term prochirality is redefined as having a purely geometric meaning, a method based on probe stereoisograms and another method based on equivalence classes (orbits) are introduced to determine prochirality and/or pro-RS-stereogenicity. Enantiotopic and RS-diastereotopic relationships appearing in probe stereoisograms are respectively used to determine prochirality and pro-RS-stereogenicity, where ‘stereoheterotopic’ relationships used in modern stereochemistry are abandoned. Alternatively, an enantiospheric orbit for specifying prochirality and an RS-enantiotropic orbit for specifying pro-RS-stereogenicity are emphasized by using coset representations and Young tableaux. The pro-R/pro-S-system is clarified to be based on pro-RS-stereogenicity and not on prochirality.

The continuous orbifold of N $$ \mathcal{N} $$ = 2 minimal model holography

For the $$ \mathcal{N} $$ = 2 Kazama-Suzuki models that appear in the duality with a higher spin theory on AdS3 it is shown that the large level limit can be interpreted as a continuous orbifold of 2N free bosons and fermions by the group U(N ). In particular, we show that the subset of coset representations that correspond to the perturbative higher spin degrees of freedom are precisely described by the untwisted sector of this U(N ) orbifold. We furthermore identify the twisted sector ground states of the orbifold with specific coset representations, and give various pieces of evidence in favour of this identification.

ChemInform Abstract: A Model for Combinatorial Organic Chemistry.

The set of coset representations, CR's, of a group G, {G(/G1), G(/G2), ..., G(/Gs)}, where G1 = {I}, Gs = G, the marks, mij of subgroup Gj on a given G(/Gi), 1 ≤ i ≤ s, and the subduction of G(/Gi) by Gj, j ≤ i, G(/Gi) ↓ Gj, are essential tools for the enumeration of stereoisomers and their classification according to their subgroup symmetry (Fujita, S. Symmetry and Combinatorial Enumeration in Chemistry; Springer−Verlag: Berlin, 1991). In this paper, each G(/Gi) is modeled by a set of colored equivalent configurations (called homomers), ℋ = {h1, h2, ..., hr}, r = |G|/|Gi|, such that a given homomer, hk, remains invariant only under all g ∈ Gi, where g is an element of symmetry. The resulting homomers generate the corresponding set of marks almost by inspection. The symmetry relations among a set ℋ can be conveniently stored in a Cayley-like diagram (Chartrand, G. Graphs as Mathematical Models; Prindle, Weber and Schmidt Incorporated: Boston, MA, 1977; Chapter 10), which is a complete digraph on r vertic...

Combinatorial Properties of Graphs and Groups of Physico-Chemical Interest

Combinatorial properties of graphs and groups of physico-chemical interest are described. A type of mathematical modeling is applied which involves "translating" algebraic expressions into graphs. The idea is applied to both graph theory and group theory. The former topic includes objects of importance in physics and chemistry such as trees, polyomino graphs, king boards, etc. Our study along these lines emphasizes nonadjacency relations, graph-generation, quasicrystals, continued fractions, fractals, and general ordering schemes of graphs. The second part of the paper considers certain colored graphs as models of several group-theoretical concepts including coset representations of groups, subduction of groups, character tables, and mark tables which are essential to the understanding of recent developments of combinatorial enumeration in chemistry.

Group-theoretical Foundations for the Concept of Mandalas as Diagrammatical Expressions for Characterizing Symmetries of Stereoisomers

Group-theoretical foundations for the concept of mandalas have been formulated algebraically and diagrammatically in order to reinforce the spread of the unit-subduced-cycle-index (USCI) approach (S. Fujita, Symmetry and Combinatorial Enumeration in Chemistry, Springer-Verlag, Berlin-Heidelberg, 1991). Thus, after the introducton of right coset representations (RCR) (H\)G and left coset representations (LCR) G(/H) for the group G and its subgroup H, a regular body of G-symmetry is defined as a diagrammatical expression for a right regular representation (C 1\)G, which is an extreme case of RCRs. The |G| substitution positions of the regular body as a reference are numbered in accord with the numbering of the elements of G and segmented into |G|/|H| of H-segments, which are governed by the RCR (H\)G. By regarding each H-segment as a substitution position, the H-segmented regular body is reduced into a reduced regular body, which can be regarded as a secondary skeleton for generating a molecule. The reference regular body (or H-segmented one) is operated by every symmetry operations of G to generate regular bodies (or H-segmented ones), which are placed on the vertices of a hypothetical regular body of G-symmetry. The resulting diagram (a nested regular body) is called a mandala (or a reduced mandala), which is a diagrammatical expression for specifying the G-symmetry of a molecule. The effect of a K-subduction on the regular bodies of a mandala (or a reduced mandala) results in the K-assemblage of the mandala (or the reduced mandala), where the resulting K-assemblies governed by the LCR G(/K) construct a |G|/|K|-membered orbit, which corresponds to a molecule of K-symmetry. The sphericity of the RCR (or the LCR) is used to characterize symmetrical properties of substitution positions and those of stereoisomers. The fixed-point vector for each mandala (or reduced mandala) in terms of row view and the number of fixed points of K-assembled mandalas (or K-assembled reduced mandalas) in terms of column view are compared to accomplish combinatorial enumeration of stereoisomers. The relationship between a mandala and a reordered multiplication table is discussed.

Sphericities of Double Cosets, Double Coset Representations, and Fujita’s Proligand Method for Combinatorial Enumeration of Stereoisomers

The concepts of double coset representations and sphericities of double cosets are proposed to characterize stereoisomerism, where double cosets are classified into three types, i.e., homospheric double cosets, enantiospheric double cosets, or hemispheric double cosets. They determine modes of substitutions (i.e., chirality fittingness), where homospheric double cosets permit achiral ligands only; enantiospheric ones permit achiral ligands or enantiomeric pairs; and hemispheric ones permit achiral and chiral ligands. The sphericities of double cosets are linked to the sphericities of cycles which are ascribed to right coset representations. Thus, each cycle is assigned to the corresponding sphericity index (a d , c d , or b d ) so as to construct a cycle indices with chirality fittingness (CI-CFs). The resulting CI-CFs are proved to be identical with CI-CFs introduced in Fujita’s proligand method (S. Fujita, Theor. Chem. Acc. 113 (2005) 73–79 and 80–86). The versatility of the CI-CFs in combinatorial enumeration of stereoisomers is demonstrated by using methane derivatives as examples, where the numbers of achiral plus chiral stereoisomers, those of achiral stereoisomers, and those of chiral stereoisomers are calculated separately by means of respective generating functions.

Complete settlement of long-standing confusion on the term ‘prochirality’ in stereochemistry. Proposal of pro- RS-stereogenicity and integrated treatment with prochirality

Long-standing confusion on the term ‘prochirality’ in stereochemistry has been analyzed by a critical review on the definitions described in the IUPAC recommendations 1996 ( Pure Appl. Chem., 1996, 68, 2193). Thereby, the confusion has been clarified to come from the fact that the scope and limitations of the terms ‘enantiotopic’, ‘diastereotopic’, and ‘stereoheterotopic’ are not so fully specified to discriminate between prochirality and prostereogenicity. Entangled situations due to the confusion have been avoided by abstracting the concept of pro- RS-stereogenicity from the conventional ‘prostereogenicity’ on the same line as the concept of RS-stereogenicity was separated from the conventional ‘stereogenicity’ (S. Fujita, J. Org. Chem., 2004, 69, 3158). This abstraction, which has been based on stereoisograms and RS-stereoisomeric groups, has provided us with a systematic approach for investigating the relationship between the prochirality and the pro- RS-stereogenicity. Thus, the prochirality and the pro- RS-stereogenicity can be discussed in terms of a common theoretical framework, that is, coset representations of RS-stereoisomeric groups, where the conventional terms on topicities are replaced by unambiguous terms on sphericities (homospheric, enantiospheric, and hemispheric) and RS-tropicities ( RS-homotropic, RS-enantiotropic, and RS-hemitropic). Moreover, the pro- RS-stereogenicity defined in the present paper has been correlated to the capability of designating pro- R/ pro- S descriptors without any ambiguity. Thereby, the long-standing confusion on the term ‘prochirality’ has been settled completely.

Graphs to chemical structures 1. Sphericity indices of cycles for stereochemical extension of P�lya?s theorem

Polya’s theorem has been concluded to be concerned with graphs, but not with chemical structures, where it is incapable of treating chiral ligands properly. In order to take account of chiral ligands along with achiral ones, coset representations (CRs) for cyclic subgroups have been examined to classify permutations of the CRs into proper and improper elements. As a result, a k-cycle contained in each permutation has been classified into an enantiospheric, homospheric, or hemispheric one. Thereby, sphericity indices of k-cycles have been defined according to the enantiospheric, homospheric, or hemispheric nature of each k-cycle. On the basis of the sphericity indices, cycle indices with chirality fittingness (CIs-CF) have been defined in place of Polya’s cycle indices. The CIs-CF have been proved to be capable of enumerating of stereoisomers with chiral and achiral ligands. Their capabilities have been confirmed by using allene derivatives as examples.

Doubly-Colored Graphs as Graphical Models for Subductions of Coset Representations, Double Cosets, and Unit Subduced Cycle Indices

The concept of doubly-colored graphs is proposed to model subductions of coset representations, double cosets, and unit subduced cycle indices, which have been mathematically formulated in coset algebraic theory developed by Fujita (Symmetry and Combinatorial Enumeration in Chemistry, Springer-Verlag, Berlin, 1991).

組み合わせ化学のグラフ的モデル化

Traditional textbooks [1–3] on chemical group theory deal with linear representations and character tables of groups, which are usually directed to modeling molecular vibrations, infrared spectroscopy, and symmetrysimplification of secular-determinants in molecular orbital calculations. On the other hand, Fujita [4] adopted an alternative approach which is based on permutation representations (coset representations) and mark tablesg, where he developed the USCI (unit-subducedcycle-index) method [4, 5], aiming at studying combinatorial enumeration in chemistry. The two approaches of chemical group theory have been compared [6] after the development of the characteristic-monomial method for combinatorial enumeration on the basis of the former approach [7, 8]. While the former approach requires conjugacy relations between elements of a group only, the latter approach requires a knowledge of all group/subgroup relations which may not be trivial to find out. Usually a group G is expressed as the set of elements of symmetry g’s which it contains:

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.

Sphericity governs both stereochemistry in a molecule and stereoisomerism among molecules.

The concept of sphericity and relevant fundamental concepts that we have proposed have produced a systematized format for comprehending stereochemical phenomena. Permutability of ligands in conventional approaches is discussed from a stereochemical point of view. After the introduction of orbits governed by coset representations, the concepts of subduction and sphericity are proposed to characterize desymmetrization processes, with a tetrahedral skeleton as an example. The stereochemistry and stereoisomerism of the resulting promolecules (molecules formulated abstractly) are discussed in terms of the concept of sphericity as a common mathematical and logical framework. Thus, these promolecules are characterized by point group and permutation group symmetry. Prochirality, stereogenicity, prostereogenicity, and relevant topics are described in terms of the concept of sphericity.

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.

The unit-subduced-cycle-index methods and the characteristic-monomial method. Their relationship as group-theoretical tools for chemical combinatorics

Among the four methods of the unit-subduced-cycle-index (USCI) approach, the subduced-cycle-index (SCI) method and the partial-cycle-index (PCI) method have been discussed by using adamantane of Td-symmetry as a probe for enumeration problems, where USCIs are derived on the basis of permutaion representations, coset representations (CRs) and marks. After the examination of the SCIs and PCIs, Polya's theorem that is a standard method of chemical combinatorics has been derived from the USCI approach. As another approach, a new method called the characteristic-monomial (CM) method has been developed by virtue of charactereistic monomials (CMs). The CMs have been derived from Q-conjugacy representations and Q-conjugacy characters, which have been related to irreducible representations and irreducible characters of the standard repertoire of chemical group theory. The two approaches have been compared to discuss group-theoretical tools for chemical combinatorics on a common basis.