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The aim of this short note is to improve some recent results of B. Grunbaum by some remarks. We use Grunbaum's notations.

The structure of the ${\ensuremath{\Gamma}}_{25}^{+}$ degenerate valence band of Ge has been investigated by a comprehensive study of the complex "quantum" resonance spectra from electric-dipole transitions in the system of anomalously spaced low-lying Landau levels produced by an applied magnetic field. This paper, the first of a series of four, is devoted to the development of a systematic theoretical spectroscopy to serve as a framework for the analysis of our experiments which will be discussed in succeeding papers. Using group-theoretical techniques, following Luttinger, we construct a generalized effective-mass Hamiltonian for holes, including the effects of elastic strain, in the full six-dimensional space of ${\ensuremath{\Gamma}}_{25}^{+}$ to encompass the spin-orbit-split-off band. The formulation in the extended space sheds considerable light on two important consequences of spin-orbit coupling, the anisotropy of the hole $g$ factor (the Luttinger $q$ term) and a new effect, the spin-dependent contribution to the valence-band deformation potentials. From the Hamiltonian for ${\ensuremath{\Gamma}}_{25}^{+}$ we project, making the split appropriate to large spin-orbit interaction, the Hamiltonian belonging to the subspace of the band edge ${\ensuremath{\Gamma}}_{8}^{+}$. We examine the nature of its eigenstates and develop a systematic scheme based on group theory for classifying the magnetic eigenstates, in terms of which selection rules for quantum transitions can be expressed in unusually clear and compact form. A formalism is presented for generating complete, "synthesized" quantum-resonance spectra starting with the eigenvalues and eigenfunctions of the effective-mass Hamiltonian. A further projection, representing the decoupling of the ${\ensuremath{\Gamma}}_{8}^{+}$ band by large uniaxial stress, is expanded to second order to evaluate the corrections to the hole effective masses and $g$ factors at finite stress. Finally, we consider the interaction between the projected spaces of the stress-decoupled band-edge states and the spin-orbit-split-off states which contributes two important shifts to the quantum-resonance effective masses: the second-order magnetic interaction and the interaction from the cross terms of strain and magnetic field.

That the factor space of a semisimple Lie group by an arithmetic subgroup has finite volume with respect to Haar measure is well known. In this paper we study results related to the converse of this theorem. In particular, under some rather weak assumptions on a semisimple Lie group G we prove that every discrete subgroup of G with a non-compact factor space of finite volume that satisfies some natural irreducibility conditions, is an arithmetic subgroup of G. In this paper we also study various results from the theory of algebraic groups and their arithmetic and discrete subgroups. In the proof of one theorem we use a construction from representation theory that is of independent interest. At the end we state some unsolved problems in the theory of discrete subgroups.

The aim of these two papers (I and II) is to try to give a unified systematic description of the different quantum mechanical realization of the unitary representations of the inhomogeneous space-time groups. Paper I, which is more general, deals with the interrelationships between the Mackey, the Wigner and the covariant realizations of induced representations, first for general groups, and then for semi-direct products. The Euclidean, Galilean and Poincare groups are treated in some detail as examples. Paper II gives a systematic treatment of the various covariant wavefunctions (Dirac, Bargmann-Wigner, etc.) that can be used to describe a single irreducible representation of the Poincare group. It is self-contained and deals first with momentum space, and then with first and second quantized fields. In both papers the treatment is index-free.

The little group of the k vector is given at each special point of the Brillouin Zone of every cubic space group, and the unitary subgroup is identified in all of the magnetic groups derived from each of these. Using the theory due to Wigner the irreducible corepresentations (" coreps ") are derived as an example in the case of magnetic groups related to Th7 (la3). the irreducible representations ("reps ") of the group of the k vector at special points of symmetry of the Brillouin Zones (BZs) of three cubic structures there have been large numbers of papers considering the same thing for many various space groups which authors have reduced as they became interested in a sub­ stance exhibiting a particular space group. And with the advent of the study of double and magnetic groups more recently the profusion of papers has in­ creased still further. It was felt that it is time that a unified treatment, in English, of all the space groups was given irrespective of their immediate ap­ plication, in the belief that, in the end, this will represent a saving of effort. The first stage in this programme is represented by a recent pair of papers dealing with the space groups PJn3m ( 0,/), Fm3m ( Oh 3 l Further material on the rest of the space groups is in prepa­ ration. In the present paper the obvious extension from the irreducible repre­ sentations of little groups for the 230 space groups to the problem of the representations of the Shubnikov or magnetic groups is considered. § 2. The group of the k vector for cubic space groups In one of the papers already referred to 2 l the group of the k vector for the points of symmetry in the BZs of the space groups Pm3m, Fm3m and lm3m were given. We use the notation which was used there for the opera­ tions of the space groups, and in Table I list the groups of the k vector for the points of symmetry for the remaining symmorphic cubic space groups, of which there are twelve. Taking for example those space groups which are subgroups of Pm3m(O,/), namely P23(T 1 ), Pm 3(T,/), P43m(T(/) and P432(0

This review is the third in the Annual Reviews of Physical Chemistry (1) during the past eight years that treats the optical properties of molecular crystals. Two other reviews (2) on this same type of subject matter, but with slightly different emphasis, have appeared recently in the literature. Besides these review articles, other recent treatises are strongly recommended to the reader. Knox's book (3) , for example, presents a much broader presentation of exciton theory and practice than this review attempts to do. For those readers who are interested in a unified approach, not just to the Frenkel exciton problem in organic crystals, but also to other kinds of excitons in other kinds of crystals, Knox's book is strongly recommended as a starting point. A condensed but rather clear discussion of many of the important aspects of space group theory and time reversal symmetry as they apply to the general energy band problem is to be found in the book by Callaway (4) . Other good sources for this type of information are the classic paper by Koster (5) and the books on applied group theory by Tinkham (6) and Mariot (7) . Craig & Walmsley (8) outline the fundamental theoretical material neces­ sary for an understanding of the subject matter presented in this review. A fairly recent paper by Davydov (9) corrects, clarifies, and extends Davydov's own classical work on the molecular exciton problem. Because of the exist­ ence of this publication, the original book in Russian ( 10) and its later English translation ( 1 1 ) , which describes Davydov's early work, are now to be valued more for their historical content and bibliographies than as a source for study of modern molecular exciton theory. Since Hochstrasser's review, a large number of new experimental and theoretical techniques have been introduced leading to many significant advances in molecular exciton theory. The field is a much more polished one than it was in 1966, and its scope is considerably broader. The field too has taken on new significance in recent years because of the partial shift of

In two previous papers (1, 2) representations of the unitary groups U4, U2 were found which described some of the properties of nucleons and electrons. In particular, the many electron wave functions were constructed from the irreducible representations of U2 restricted to the proper orthochronous Lorentz group Lp. In this paper the irreducible representations of U4 found in (1) will be shown to be also irreducible representations of the complete homogeneous Lorentz group L0 and the techniques of matrix contraction employed in (2) will be used to find the precise form of the matrices of the infinitesimal ring.

Introduction. The problem of constructing all the finite metabelian groups (that is, groups with an abelian commutator subgroup) is fundamentally settled by the Schreier theory of group extensions. In fact, a metabelian group may be considered as the extension of an abelian group 2a by an abelian group a; this is the simplest nontrivial case of a group extension. According to Schreier's theory, an arbitrary group 5 which is the extension of 2 by a is obtained by the following procedure: First, we have to find an automorphism group of 2f which is the homomorphic image of a, that is, if ais the automorphism corresponding to a Ea, then (AB) =A Ba, (A)a =Aa 5=A " for every A, B GX, o, rEG. Secondly, we have to find a factor system Ca, in A, satisfying CaT, CaT= C7,P for every pi o, ir (see [18, p. 90])(l). If Sa is a symbol denoting a certain representative of oin 0, then the relations SaAS,i =Aa, S,S, = C?,7Sa7 uniquely determine an abstract group 5 with the required properties. Unfortunately, the general formulation of the Schreier theory does not indicate (except in the most trivial cases) how to determine and specify the automorphisms and factor systems so that each 5 shall be obtained in one and only one way. The invariant characterization of solvable groups, even in the relatively simple case of metabelian groups, still remains one of the most important and most difficult problems of abstract group theory. At the present, it seems that the problem can be successfully approached only if we impose certain restrictions upon the family of groups to be determined. In a recent paper [12] I have determined all the groups (5 which have an abelian invariant subgroup 21 of the type (p, * *, p) such that 9/2 be cyclic. In the present paper a more extensive class of metabelian groups will be determined and completely characterized by numerical invariants. Whereas no restriction will be imposed upon the structure of the abelian invariant subgroup X, it is assumed that 9/f is cyclic and its order n is not divisible by the square of any prime number which divides the order of 2W. In particular, the latter condition is fulfilled if either n is squarefree, or if n and the order of 2a are relatively prime. Among the more important cases included in the above category (others will be mentioned in part 3) perhaps the most notable is the case of p-groups

We give in this paper a detailed and unified description of the procedures which must be followed for all cases of physical interest to go from the wave functions carrying an irreducible representation of the restricted Poincare group, to objects having a manifestly covariant behaviour. As well known, for instance for the case of spin 1/2 massive particles, this amounts to performing the inverse of a Foldy-Wouthuysen transformation. Before coming to the main subject of the paper, the construction of the irreducible representations of the restricted and extended Poincare group are reviewed. Particular emphasis is given to the physical meaning of the representations and of the invariants which characterize them. After this, we derive manifestly covariant quantities from the wave functions of the irreducible representations of the restricted group for spin 0. 1/2 and 1, massive and massless elementary systems. The corresponding relativistically invariant equations, i.e. the Klein-Gordon, Dirac, Proca, Weyl and Maxwell equations are then derived. The transformations properties under the transformations of the extended group are also extensively discussed.

This paper solves the following problems. Let G be a finitely presented group given by generators g 1 , g 2 , …, g k and relations R 1 =R 2 =…=R t =I where R i (i= 1, 2, …, t ) are give words in g 1 , g 2 , …, g k and their inverses. Let H be a subgroup of G of finite index, which is generated by a finite set of words, h 1 , h 2 , …, h r in the generators of G and their inverses. Find a set of representatives σ 1 , σ 2 , …, σ 3 of the right cosets of H , and furthermore, given a word W in the generators of G and their inverses, express W in the form W=W * σ i where W * is a word in h 1 , h 2 , … h r and their inverses. The solution to this problem enables one to solve various interesting problems in group theory. The computation is extremely practical and may be programed easily on a high-speed computing machine. It may be pointed out here that this paper extends the results of [3] in that it gives a systematic procedure for expressing the Schreier generators of a subgroup as words in the originally given generators. The process was used in an informal way as far back as 1960, but at that time machine computation was not envisaged.

The purpose of this series of papers is to carry out systematic calculations in nuclear structure in the 2s-1d shell, considering a model interaction which is a linear combination of a spin-orbit coupling (Ws.o.) plus a quadrupole-quadrupole (Q2) and an orbital pairing (P) term, the last two terms taking into account, respectively, the long and short-range correlations of the residual central interaction. In this paper we discuss methods of group theory that would allow us to classify the states in relation with a chain of subgroups in such a way that each of the interactions Q2, P and Ws.o. becomes diagonal, and then we discuss how to obtain transformation brackets that take us from states classified by one chain of subgroups to states classified by another, so that it is possible to obtain the matrix elements for the model interaction. In the next papers of this series we shall apply the present techniques to specific nuclei, starting with those of A=18 and A=20.

In a previous paper, we showed how wavefunctions which transform in a relativistic manner in configuration space can be expanded in terms of amplitudes, which for nonzero mass transform like the wavefunctions for irreducible representations of the proper, orthochronous, inhomogeneous Lorentz group. A simple algorithm was given to obtain the expansion. In the present paper, we extend the results to include zero‐mass amplitudes. It is shown that for wavefunctions which are required to transform under the homogeneous Lorentz group such that the matrices which involve the spinor indices are finite dimensional, the zero‐mass amplitudes transform under nonunitary representations of the inhomogeneous Lorentz group. However, it is possible to split up each such nonunitary representation into a part which corresponds to a unitary representation for finite spin and into a part which corresponds to an unphysical change of wavefunction. As examples of the technique, we consider wavefunctions which transform as an ant...

In this paper an attempt is made to define the class of equivalent reference frames in special relativity in terms of a purely kinematical characterization of the notion of free material point and using homogeneity of space and time, isotropy of space, and unidirectional time flow. The problem of finding the possible forms of the relativity group and the significance of a ``space‐time homogeneity axiom'' which was used in previous papers are discussed.

A group theoretical method to deal with the harmonic vibrations of a general crystalline lattice is indicated in this paper. Sir C. V. Raman’s theory of lattice vibrations3 is the basis of the work and it is indicated how the normal modes, frequencies, their degeneracies, etc., can be rigorously derived. Applications of this work to specific problems are given in another paper in this symposium.