Visualizing the C2h character table: an intuitive guide through the Li2CO3 crystal structure

SpaceGroupIrep: A package for irreducible representations of space group

Generation of molecular symmetry orbitals for the point and double groups

This chapter defines the representations of finite groups and considers successively the cases of crystallographic point groups in one, two and three dimensions and of space groups. It introduces the notions of irreducible representations. The character tables and irreducible representations are given for the 32 crystallographic point groups in three dimensions. The transformation properties of tensors are also considered. In the last section, the accompanying software Tenχar is introduced. This software can be used to determine the irreducible representations of finite point groups in three dimensions (the 32 crystallographic groups and the groups of the quasicrystalline phases) and the independent components of a tensor of any rank for each of these groups.

Chapter 4 - Group Theory: Matrix Representation and Character Tables

This chapter introduces group theory and the rules governing mathematical groups, of which point groups and space groups are examples. Group multiplication tables and their uses are described with examples, and subgroups and cosets are considered briefly. Similarity transformations are used to show the equivalence, or otherwise, of symmetry operations. Character tables are introduced and explained, and the procedures for creating representations are described. Transformation properties are discussed, and irreducible representations and their notations are examined, leading to a description of the theorems on orthogonality. The construction of character tables is explained, together with the applications of direct products.

The definition of a character has been given in Chapter 3. As will be demonstrated in Chapter 5, character tables are necessary for the determination of the selection rules that govern which bands will appear in the infrared and Raman spectra of various molecules and also for the determination of the number of fundamentals of each type of vibration. Table 4–1 shows a typical character table, that for the C 2v point group, and Fig. 4-1 diagrammatically illustrates the significance of the various parts for the character table for C 3v. The character table classifies the displacements of the atoms of molecules from their equilibrium positions according to the irreducible representation of the symmetry group. The first column of the character table lists the types of representations, or species of vibrations, possible for the given point group. The most symmetrical species are placed near the top of the table, and the least symmetrical species near the bottom. The symmetry classes pertinent to the point group form the column headings.

Multiple antennas can greatly increase the data rate and reliability of a wireless communication link in a fading environment. Their success, however, depends on the design of cedes that achieve these promises. It is well known that unitary matrices can be used to design differentially modulated space-time codes. These codes have a particularly efficient description if they form a finite group under matrix multiplication. We show how to compute the parameters of such groups crucial for their use as space-time codes, using only the character table of the group. Since character tables for many groups are known and tabulated, this method could be used to quickly test, for a given group, which of its irreducible representations can be used to design good unitary space-time codes. We demonstrate our method by computing the eigenvalues of all the irreducible representations of the special linear group SL/sub 2/(F/sub q/) over a finite prime field F/sub q/ of odd characteristic, and study in detail the performance of a particular eight-dimensional representation of SL/sub 2/(F/sub 17/).

The properties of the complex band structure in diamond-type crystals are studied. The symmetry of various surface states ({100}, {110}, {111} faces) is discussed both with and without spin-orbit coupling. Character tables and compatibility relations are obtained for the various irreducible representations of the surface space groups and compatibility relations between these representations and those of the three-dimensional space group are also discussed. The electronic band structure is presented for complex k vectors in the [100] and [111] directions.

In the analysis of spin structures a “natural” point of view looks for the set of symmetry operations which leave the magnetic structure invariant and has led to the development of magnetic or Shubnikov groups. A second point of view presented here simply asks for the transformation properties of a magnetic structure under the classical symmetry operations of the 230 conventional space groups and allows one to assign irreducible representations of the actual space group to all known magnetic structures. The superiority of representation theory over symmetry invariance under Shubnikov groups is already demonstrated by the fact proven here that the only invariant magnetic structures describable by magnetic groups belong to real one-dimensional representations of the 230 space groups. Representation theory, on the other hand, is richer because the number of representations is infinite, i.e., it can deal not only with magnetic structures belonging to one-dimensional real representations, but also with those belonging to one-dimensional complex and even to two-dimensional and three-dimensional representations associated with any k vector in or on the first Brillouin zone. We generate from the transformation matrices of the spins a representation Γ of the space group which is reducible. We find the basis vectors of the irreducible representations contained in Γ. These basis vectors are linear combinations of the spins and describe the structure. The method is first applied to the k = 0 case, where magnetic and chemical cells are identical and then extended to the case where magnetic and chemical cells are different (k≠O) with special emphasis on k vectors lying on the surface of the first Brillouin zone in nonsymmorphic space groups. As a specific example, we consider several methods of finding the two-dimensional irreducible representations and its basis vectors associated with k = 12b2 = [0120] in space group Pbnm - (D2h16). We illustrate the physical context of representation theory by constructing an effective spin Hamiltonian H invariant under the crystallographic space group and under spin reversal. H is even in the spins and automatically invariant under the (isomorphous) magnetic group. We show by the example of CoO that the invariants in H, formed with the help of basis vectors, give information on the nature of spin coupling as, for instance, isotropic (Heisenberg-Néel) coupling, vectorial (Dzialoshinski-Moriya) and anisotropic symmetric couplings. Magnetic structures, cited in the text to show the implications of the representation theory of space groups are ErFeO3, ErCrO3, TbFeO3, TbCrO3, DyCrO3, YFeO3, V2CaO4, β-CoSO4, Er2O3, CoO, and RMn2O5 (R = Bi, Y, or rare earth). Representation theory of magnetic groups must be considered when the Hamiltonian contains terms which are odd in the spins. The case may occur when the magnetic energy is coupled with other forms of energy as, for instance, in the field of magnetoelectricity. Here again representation theory correctly predicts the couplings between magnetic and electric polarizations as shown on LiCoPO4 and (previously) on FeGaO3. A full account of this work has been submitted elsewhere.

In the analysis of spin structures a `natural' point of view looks for the set of symmetry operations which leave the magnetic structure invariant and has led to the development of magnetic or Shubnikov groups. A second point of view presented here simply asks for the transformation properties of a magnetic structure under the classical symmetry operations of the 230 conventional space groups and allows one to assign irreducible representations of the actual space group to all known magnetic structures. The superiority of representation theory over symmetry invariance under Shubnikov groups is already demonstrated by the fact proven here that the only invariant magnetic structures describable by magnetic groups belong to real one-dimensional representations of the 230 space groups. Representation theory on the other hand is richer because the number of representations is infinite, i.e. it can deal not only with magnetic structures belonging to one-dimensional real representations, but also with those belonging to one-dimensional complex and even to two-dimensional and three-dimensional representations associated with any k vector in or on the first Brillouin zone. We generate from the transformation matrices of the spins a representation Γ of the space group which is reducible. We find the basis vectors of the irreducible representations contained in Γ. The basis vectors are linear combinations of the spins and describe the structure. The method is first applied to the k = 0 case where magnetic and chemical cells are identical and then extended to the case where magnetic and chemical cells are different (k ≠ 0) with special emphasis on k vectors lying on the surface of the first Brillouin zone in non-symmorphic space groups. As a specific example we consider several methods of finding the two-dimensional irreducible representations and its basis vectors associated with k = ½ b2 = [0½0] in space group Pbnm (D162h). We illustrate the physical context of representation theory by constructing an effective spin Hamiltonian H invariant under the crystallographic space group and under spin reversal. H is even in the spins and automatically invariant under the (isomorphous) magnetic group. We show by the example of CoO that the invariants in H, formed with the help of basis vectors, give information on the nature of spin coupling as for instance isotropic (Heisenberg–Néel) coupling, vectorial (Dzialoshinski–Moriya) and anisotropic symmetric couplings. Magnetic structures, cited in the text to show the implications of the representation theory of space groups are ErFeO3, ErCrO3, TbFeO3, TbCrO3, DyCrO3, YFeO3, V2CaO4, β-CoSO4, Er2O3, CoO and RMn2O5 (R = Bi, Y or rare earth). Representation theory of magnetic groups must be considered when the Hamiltonian contains terms which are odd in the spins. The case may occur when the magnetic energy is coupled with other forms of energy as for instance in the field of magneto-electricity. Here again representation theory correctly predicts the couplings between magnetic and electric polarizations as shown on LiCoPO4 and (previously) on FeGaO3.

We attempt to give a complete description of the ‘exceptional’ finite subgroups Σ(36 × 3), Σ(72 × 3) and Σ(216 × 3) of SU(3), with the aim to make them amenable to model building for fermion masses and mixing. The information on these groups which we derive contains conjugacy classes, proper normal subgroups, irreducible representations, character tables and tensor products of their three-dimensional irreducible representations. We show that, for these three exceptional groups, usage of their principal series, i.e. ascending chains of normal subgroups, greatly facilitates the computations and illuminates the relationship between the groups. As a preparation and testing ground for the usage of principal series, we study first the dihedral-like groups Δ(27) and Δ(54) because both are members of the principal series of the three groups discussed in the paper.

The character table of the fully nonrigid water pentamer, (H2O)5, is derived for the first time. The group of all feasible permutations is the wreath product group S5[S2] and it consists of 3840 operations divided into 36 conjugacy classes and irreducible representations. We have shown that the full character table can be constructed using elegant matrix type generator algebra. The character table has been applied to the water pentamer by obtaining the nuclear spin statistical weights of the rovibronic levels and tunneling splittings of the fully nonrigid pentamer. We have also obtained the statistical weights and tunneling splittings of a semirigid deuterated pentamer that exhibits pseudorotation with an averaged C5h (G10) symmetry used in the assignment of vibration-rotation−tunneling spectra. It is also shown that the previously considered group G320 for water pentamer of feasible permutations is a subgroup of the full group and is the direct product of wreath product C5[S2] and the inversion group. The correlation tables have been constructed for the semirigid (G10) to nonrigid (G3840) groups for the rotational levels and tunneling levels. The nuclear spin statistical weights have also been derived for both the limits and through the use of subduced representations the corresponding information can be obtained for G(320) as well from G(3840).

It is demonstrated that the accepted procedure of constructing character tables by making use of the known rules derived from The Great Orthogonality Theorem about the irreducible representations and their characters, can give more than one table as character table for some symmetry point groups. All these tables adhere perfectly well to the rules. The limitation to the known method is discussed. The procedure to obtain the correct table is indicated .

Nested wreath product groups arise from looped or recursive structures that contain repeated copies of the same structure one within the other. Phylogeny trees in biology, Cayley trees, Bethe lattices, NMR graphs of non-rigid molecules, ammoniated ammonium ions are all examples of structures that exhibit such nested wreath product automorphism groups. We show that the conjugacy classes, irreducible representations and character tables of these nested group structures can be generated using multinomial generating functions cast in terms of matrix types that can be simplified into generalized cycle type polynomials. The nested wreath product groups rapidly increase in orders, for example, a simple wreath product group \(\hbox {S}_{7}[\hbox {S}_{7}]\) consists of \((7!)^{8}\) or \(4.1633\times 10^{23}\) operations, 481,890 conjugacy classes, spanning a 481,891 \(\times \) 481,891 character table that would occupy 232,217,972 pages. We have obtained powerful recursive relations for the conjugacy classes, character tables and the orders of various conjugacy cases of any nested wreath product \(\{[\hbox {S}_{\mathrm{n}}]\}^{\mathrm{m}}\) or \(\hbox {S}_{\mathrm{n}}[\hbox {S}_{\mathrm{n}}[\hbox {S}_{\mathrm{n}}{\ldots }.[\hbox {S}_{\mathrm{n}}]]{\ldots }.]\) with order \(\left( {n!}\right) ^{a_m},\,\hbox {a}_{\mathrm{m}}=(\hbox {n}^{\mathrm{m}}-1)/(\hbox {n}-1)\). We have obtained the character tables of phylogenetic trees of any order, character tables of Cayley trees of degrees 3 and 4 and for Cayley trees of larger degrees, we have derived exact analytical expressions for the conjugacy classes and IRs for up to \(\{[\hbox {S}_{7}]\}^{\mathrm{m}}\) with order \((7!)^{137257}\) for \(\hbox {m}=7\). Applications to colorings of phylogenic trees in biology are considered.

Character tables of finite groups and closely related commutative algebras have been investigated recently using new perspectives arising from the AdS/CFT correspondence and low-dimensional topological quantum field theories. Two important elements in these new perspectives are physically motivated definitions of quantum complexity for the algebras and a notion of row-column duality. These elements are encoded in properties of the character table of a group G and the associated algebras, notably the centre of the group algebra and the fusion algebra of irreducible representations of the group. Motivated by these developments, we define a notion of generator complexity for commutative Frobenius algebras with combinatorial bases. In the context of finite groups, this gives rise to row and column versions of generator complexity for character tables. We investigate the relation between these complexities under the exchange of rows and columns. We observe regularities that arise in the statistical averages over small character tables and propose corresponding conjectures for arbitrarily large character tables.