Dominant representations and a markaracter table for a group of finite order

Summary The concept of markaracter is proposed to discuss marks and characters for a group of finite order on a common basis. Thus, we consider a non-redundant set of dominant subgroups and a non-redundant set of dominant representations (SDR), where coset representations concerning cyclic subgroups are named dominant representations (DRs). The numbers of fixed points corresponding to each DR are collected to form a row vecter called a dominant markaracter (mark-character). Such dominant markaracters for the SDR are collected as a markaracter table. The markaracter table is related to a subdominant markaracter table of its subgroup so that the corresponding row of the former table is constructed from the latter. The data of the markaracter table are in turn used to construct a character table of the group, after each character is regarded as a markaracter and transformed into a multiplicity vector. The concept of orbit index is proposed to classify multiplicity vectors; thus, the orbit index of each DR is proved to be equal to one, while that corresonding to an irreducible representation is equal to zero.     

Subduction of Q-conjugacy representations and characteristic monomials for combinatorial enumeration

A new method of combinatorial enumeration is presented. The subduction of Q-conjugacy representations gives a characteristic subduction table and a characteristic monomial table. A cycle index is defined on the basis of such monomials and used for combinatorial enumeration of isomers. Received: 10 October 1997 / Accepted: 13 February 1998 / Published online: 17 June 1998     

Möbius function and characteristic monomials for combinatorial enumeration

After the definitions of amplified representations and number-theoretical vectors, the markaracter table of a cyclic subgroup is converted into the corresponding Q-conjugacy character table. The conversion is shown to necessitate an interconversion matrix that contains M?bius functions as elements. Since the interconversion matrix gives characteristic monomials for cyclic groups, all the powers appearing in each of the characteristic monomials are shown to be integers. Characteristic monomials for finite groups are then built up by starting from those of cyclic groups. This procedure clarifies the fact that all the powers appearing in each characteristic monomial for finite groups are integers. The relationship between characteristic monomial tables and unit-subduced-cycle-index tables is discussed with respect to their application to isomer enumeration. Received: 15 July 1998 / Accepted: 16 December 1998 / Published online: 16 March 1999     

Characteristic monomials with chirality fittingness for combinatorial enumeration of isomers with chiral and achiral ligands

A new method of combinatorial enumeration based on characteristic monomials with chirality fittingness (CM-CFs) has been proposed in order to enumerate isomers with chiral ligands as well as with achiral ones. The CM-CFs have been defined as monomials that consist of three kinds of dummy variables in light of the subduction of the Q-conjugacy representations for chiral and achiral cyclic groups. A procedure of calculating CM-CFs for cyclic groups and finite groups has been discribed so as to tabulate them as CM-CF tables. Then the CM-CF method has been applied to the enumeration of isomers with achiral ligands as well as chiral ones.     

Subduction of coset representations. An application to enumeration of chemical structures with achiral and chiral ligands

Molecules derived from a parent skeleton are enumerated where both achiral ligands as well as chiral ligands are allowed. Chirality fittingness of an orbit is proposed in order to permit chiral ligands. The enumeration is conducted with and without consideration of obligatory minimum valency (OMV). The effect of the OMV is formulated by assigning different weights to the respective orbits of the parent skeleton. The importance of coset representations and their subduction by subgroups is discussed. The subduced representations are classified into three classes through their chirality fittingness, which determines the mode of substitution with chiral and achiral ligands. Several novel concepts such as a unit subduced cycle index and a subduced cycle index are given in general forms.     

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.     

Generalization of the Harary–Palmer power group theorem to all irreducible representations of object and color groups- color combinatorial group theory

The Harary–Palmer classic power group enumeration theorem applies to a group G acting on a set D of objects such as vertices, edges, faces and simultaneously with a group H acting on another set R of colors such that the power group $\hbox {H}^\mathrm{G}$ acts on the set $\hbox {R}^\mathrm{D}$ of all functions from D to R. In this paper we show for the first time that the power group enumeration can be generalized to all irreducible representations of the object group G of D and also all irreducible representations of the color group H of R. We have also provided interpretation of various power group generating functions for different irreducible representations in the context of color symmetry group theory. Special cases of the power group enumeration with all irreducible representations of G keeping the color group representation fixed to the totally symmetric representation are shown to have important enumerative combinatorial applications in a number of problems of chemistry, physics and biology that involve color symmetry (color inversions) such as magnetism, neutron imaging, NMR, ESR spectroscopy, catalytic functions of non-rigid disordered proteins, and quantum chromodynamics of strong interactions of fundamental particles.     

Enumeration of organic reactions by counting substructures of imaginary transition structures. Importance of orbits governed by coset representations

Two methods for the enumeration of organic reactions are presented in order to take obligatory minimum valencies of a given skeleton into consideration. The first method is a generalization of Pólya's theorem, in which the transitivity of the positions of the skeletons is explicitly considered. Thus, a permutation representation acting on the positions is reduced into cosec representations (CRs). In accord with this reduction, unit cycle indices derived from the CRs construct a generalized cycle index. The second method is based on the subduction of the coset representations. This contains useful concepts such as unit subduced cycle indices and subduced cycle index that afford a new type of generating functions.     

Multinomial Combinatorial Group Representations of the Octahedral and Cubic Symmetries

We consider the full multinomial combinatorics of all irreducible representations of the octahedral (cubic) symmetry as a function of partitions for vertex, face and edge colorings. Full combinatorial tables for all irreducible representations and all multinomial partitions are constructed. These enumerations constitute multinomial expansions of character-based cycle index polynomials, and grow in combinatorial complexity as a function of edge or vertex coloring partitions.     

Subduction of coset representations

Summary Enumerations of compounds based on a parent skeleton with and without the influence of obligatory minimum valency (OMV) are reported. The effect of the OMV is formulated by assigning different weights to the respective orbits of the parent skeleton. This type of enumeration requires introduction of several new concepts that are derived from the subduction of coset representations, e.g., a unit subduced cycle index, a subduced cycle index and the number of suborbits.     

Enumeration of digraphs with a given automorphism group

Four methods are described for enumerating digraphs with a given automorphism group: (1) a generating-function method based on subduced cycle indices, (2) a generating-function method based on partial cycle indices, (3) a method based on the elementary superposition theorem, and (4) a method based on the partial superposition theorem. All of these methods are based on the concept of unit subduced cycle indices and construct a set of versatile tools for combinatorial enumeration. They are applied to the enumeration of five-vertex digraphs with a given automorphism group. The table of marks and its inverse for the symmetric group of degree 5 are recalled. The table of USCIs of this roup is obtained.Dedicated to Professor Frank Harary.     

Reduced cycle indices and their applications in enumeration of NMR signals and equivalence classes

An efficient technique is formulated based on a polynomial structure which we call the reduced cycle index for the enumeration of equivalence classes and NMR signals under group action. The reduced cycle indices are shown to be the cycle index polynomials of a subset of significantly smaller order of the point group of the molecule. Thus, the reduced cycle indices are much simpler and their use leads to a considerable reduction in tile computation of the generating functions from the cycle indices and irreducible representations contained in a set.Alfred P. Sloan Fellow; Camille and Henry Dreyfus Teacher-Scholar.     

Rational groups and integer-valued characters of Thompson group <Emphasis Type="Italic">Th</Emphasis>

According to the main result of W. Feit and G. M. Seitz (Illinois J Math 33(1):103–131, 1988), the Thompson group Th is non-rational or unmatured group (S. Fujita in Bull Chem Soc Jpn 71:2071–2080, 1998). Using the concept of markaracter tables proposed by S. Fujita (Bull Chem Soc Jpn 71:1587–1596, 1998), we are able to obtain tables of integer-valued characters for finite unmatured groups. In this paper, the integer-valued character for Thompson group is successfully derived for the first time.     

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, m(ij) of subgroup Gj on a given G(/Gi), 1 < or = i < or = s, and the subduction of G(/Gi) by Gj, j < or = i, G(/Gi) precipitates to 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, h(k), remains invariant only under all g epsilon 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 vertices so that an arc from vertex v(i) to vertex v(j) is colored with the set Sij of symmetry elements such that h(i)[g(if)]-->h(j),g(ij) epsilon Sij. In addition, each vertex, v(i), is associated with a loop that is colored with a set Sii so that g(ii) epsilon Sii stabilizes h(i). A Cayley-like diagram of a given CR, G[G(/Gi)], leads to graphical generation of G(/Gi) precipitates to Gj for all values of j and also to all m(ij)'s. Several group-theoretical results are rederived and/or became more envisagable through this modeling. The approach is examplified using C2, C3, D2, T, and D3 point groups and is applied to trishomocubane, a molecule that belongs to the D3 point group.     

Chemical graph theory. IV. On the cyclic polynomial

The definition of the cyclic polynomial of conjugated hydrocarbons is offered. The combinatorial characteristics of this polynomial are investigated. The most important property of the cyclic polynomial is that it can be used for enumeration of conjugated circuits.     

Tables of subgroups,site groups and interchange groups of symmetry point groups and the construction of SALCs

A table listing subgroups and supergroups of 43 chemically important symmetry point groups is presented. Other tables give the site groups and the corresponding interchange groups for those point groups which are of finite order. It is shown how the tables may be used to facilitate the structure determination of metal-ligand and related compounds from spectroscopic data and how the construction of symmetry adapted linear combinations. SALCs, may be simplified. The tables also may be used in factor group analysis.     

The enumeration of electron-rich and electron-poor polyhedral clusters

We present as dual processes the capping of closed triangulated polyhedra with apical atoms and the making of holes in such polyhedra either by puncture of the surface or by excision of atoms and their edges. These processes are shown to generate stable chemical species containing respectively less or more than 2n + 2 skeletal electrons. The former species are designated as electron-poor whereas the latter are called electron-rich. Pólya's enumeration method is used to enumerate the distinct ways of capping and excising the closed, triangulated polyhedra to yield systems containing from four to twelve vertices. For the enumeration of cappings the appropriate cycle index is that of the dual of the polyhedron being capped, whilst for the enumeration of the excisions the cycle index is that of the polyhedron being excised.     

Clebsch–Gordan coefficients for the group chains O ⊃ T ⊃ D2 and Td ⊃ T ⊃D2 by the method of projective representations

The spinor representations for the groups forming the chains O ? T ? D ₂ and T _d ? T ? D ₂ are constructed as projective representations. The Clebsch–Gordan coefficients are then calculated using a standard method. Projective factor systems, irreducible representations, canonical bases, and tables of Clebsch–Gordan coefficients are given. The subduction from O to D ₃ is discussed.     

The USCI approach and elementary superposition for combinatorial enumeration

Summary The elementary superposition theorems are presented for enumerating chemical compounds that contain achiral and chiral ligands. Subduced cycle indices (SCI-CF), partial cycle indices (PCI-CF), and cycle indices (CI-CF) with chirality fittingness are defined by starting from unit subduced cycle indices with chirality fittingness (USCI-CF). All of these indices afford generating functions that are proved to be applicable to combinatorial enumeration. In addition, the concept of elementary superposition with and without chirality fittingness is proposed to provide the elementary superposition theorems. These theorems provide us with a new methodology of enumerating compounds, in which the numbers of isomers are obtained without relying on generating functions and are itemized with respect to molecular formulas (weights) and symmetries. The operation is defined on the basis of the elementary superposition. Thereby, we derive superposition theorems concerning the PCI-CFs and the CI-CFs. These are applicable to combinatorial enumeration.     

Computations of Colorings 7D-Hypercube's Hyperplanes for All Irreducible Representations

In the present study, we compute and enumerate the colorings of 7D-hypercube for all of its hyperplanes (q = 1–7) for all 110 irreducible representations (IRs) of the seventh-dimensional hyperoctahedral group consisting of 645,120 symmetry operations. The computations of colorings of the 7D-hypercube are motivated by a number of chemical and biological applications such as the 7D-hypercube representation of the periodic table, hypercube representations of water heptamer clusters, genetic regulatory networks, isomerization graphs, massively large data representations, and so forth. We have employed the Möbius inversion technique combined with generalized character cycle indices for 110 IRs to compute the generating functions for colorings of seven different types of hyperplanes of the 7D-hypercube varying from the vertices (q = 7) to hexeracts (q = 1) of the 7D-hypercube. Explicit computed tables are provided for 110 IRs from q = 1 to q = 7 for the 7D-hypercube. © 2019 Wiley Periodicals, Inc.