Search

We describe a general model for the free energy function for a homogeneous medium of mutually interacting molecules, based on the formalism for a biaxial nematic liquid crystal set out by Katriel {\em et al.} (1986) in an influential paper in {\em Liquid Crystals} {\bf 1} and subsequently called the KKLS formalism. The free energy is expressed as the sum of an entropy term and an interaction (Hamiltonian) term. Using the language of group representation theory we identify the order parameters as averaged components of a linear transformation, and characterise the full symmetry group of the entropy term in the liquid crystal context as a wreath product $SO(3)\wr Z_2$. The symmetry-breaking role of the Hamiltonian, pointed out by Katriel {\em et al.}, is here made explicit in terms of centre manifold reduction at bifurcation from isotropy. We use tools and methods of equivariant singularity theory to reduce the bifurcation study to that of a $D_3\,$-invariant function on ${\bf R}^2$, ubiquitous in liquid crystal theory, and to describe the 'universal' bifurcation geometry in terms of the superposition of a familiar swallowtail controlling uniaxial equilibria and another less familiar surface controlling biaxial equilibria. In principle this provides a template for {\em all} nematic liquid crystal phase transitions close to isotropy, although further work is needed to identify the absolute minima that are the critical points representing stable phases., 74 pages, 17 figures : submitted to Nonlinearity

Group representations play a central role in theoretical physics. In particular, in quantum mechanics unitary --- or, in general, projective unitary --- representations implement the action of an abstract symmetry group on physical states and observables. More specifically, a major role is played by the so-called square integrable representations. Indeed, the properties of these representations are fundamental in the definition of certain families of generalized coherent states, in the phase-space formulation of quantum mechanics and the associated star product formalism, in the definition of an interesting notion of function of quantum positive type, and in some recent applications to the theory of open quantum systems and to quantum information., 13 pages

We build systems of imprimitivity (SI) in the context of quantum walks and provide geometric constructions for their configuration space. We consider three systems, an evolution of unitaries from the group SO3 on a low dimensional de Sitter space where the walk happens on the dual of SO3, standard quantum walk whose SI live on the orbits of stabilizer subgroups (little groups) of semidirect products describing the symmetries of 1+1 spacetime, and automorphisms (walks are specific automorphisms) on distant-transitive graphs as application of the constructions., Comment: 16 pages, 3 figures

Thinking is the development of ideas and concepts within a person caused by the process of establishing relationships between parts of the information stored in a person. In the early stages, the object of thought is accepted by the senses, processed in the brain into concepts / ideas, and expressed into a representation. Elementary students who are generally in a concrete operational stage desperately need this representation to facilitate their understanding. This research is a qualitative research with the subject of 12 students grouped into 3 groups, namely: (1) group representation in the usual way; (2) representation group with percent; (3) representation group by degrees. This research is intended to describe the thinking process of students in representing images to symbols on fractional materials. The results show that the representations of these three groups are essentially similar. All of them follow the APOS process (action, process, object and schema). The difference only occurs in group 1, where group 1 tends to think procedurally, while groups 2 and 3 think analytically.

We describe symbolic representations of the extended real line based on the dynamical systems consisting of Mobius transformations. The representations can be extended to the group of real Mobius transformations.

In this paper, we study homogeneous complex Finsler spaces. We first prove that each homogeneous complex Finsler space can be written as a coset space of a Lie group with an invariant complex structure as well as an invariant complex Finsler metric. We then introduce the notion of Minkowski representations of Lie groups and Lie algebras to give a complete algebraic description for such spaces. Finally, we study symmetric complex Finsler spaces and obtain a complete classification of such spaces.

We show that there exists a V-admissible-type subspace in the carrier space of the Wigner representation of the Poincar? group in 1+3 dimensions. This subspace is built up from an orthonormal set of Sturmian-type functions which verify naturally the assumption of rotational invariance under which relativistic coherent states frames were previously obtained. Then, we propose an extension of the concept of V-admissibility for the class of semi-direct product groups, which takes into account the relativity groups which were not covered by the existing approaches. In the light of this extension, the square-integrability modulo an appropriate subgroup of the Wigner representation of the group of Poincar? is revisited and some physical implications are discussed.

The boson calculus formalism is used to construct realizations of basis states of irreducible representations of unitary groups taking as a paradigm the interacting boson models of atomic nuclei. These realizations, together with a theorem on plethysms for obtaining branching rules, allowed us to obtain a dimension formula for reduced plethysms.

The classical solutions to higher dimensional Yang--Mills (YM) systems, which are integral parts of higher dimensional Einstein--YM (EYM) systems, are studied. These are the gravity decoupling limits of the fully gravitating EYM solutions. In odd spacetime dimensions, depending on the choice of gauge group, these are either topologically stable or unstable. Both cases are analysed, the latter numerically only. In even spacetime dimensions they are always unstable, describing saddle points of the energy, and can be described as {\it sphalerons}. This instability is analysed by constructing the noncontractible loops and calculating the Chern--Simons (CS) charges, and also perturbatively by numerically constructing the negative modes. This study is restricted to the simplest YM system in spacetime dimensions $d=6,7,8$, which is amply illustrative of the generic case., Comment: 16 pages, 3 figures ; comments added, to appear in J. Phys. A

According to the group representation theory, we derive the character formulae of representation-matrices of the physical property tensors for the one-dimensional (1D) quasicrystals. Based on this, we have calculated the numbers of independent components of representation-matrices for thermal expansion coefficient tensors, piezoelectric coefficient tensors and elastic constant tensors under 31 point-groups for the 1D quasicrystals. Moreover, we have deduced the particular matrix forms of these tensors under the 31 point-groups. This is an important complement of quasicrystal physical property.

We find that the coherent state projection operator representation of the two-mode squeezing operator constitutes a loyal group representation of symplectic group, which is a remarkable property of the coherent state. As a consequence, the resultant effect of successively applying two-mode squeezing operators are equivalent to a single squeezing in the two-mode Fock space. Generalization of this property to the 2n-mode case is also discussed.

Self-assembled quantum dots possess an intrinsic geometric symmetry. Applying group representation theory, we systematically analyze the symmetric properties of the bound states for ideal pyramid quantum dots, which neglect band mixing and strain effects. We label each bound state by its symmetry group's corresponding irreducible representation and define a concept called the quantum dots' symmetry category. A class of quantum dots with the same irreducible representation sequence of bound states are characterized as belonging to a specific symmetry category. This category concept generally describes the symmetric order of Hilbert space or wavefunction space. We clearly identify the connection between the symmetry category and the geometry of quantum dots by the symmetry category graph or map. The symmetry category change or transition corresponds to an accidental degeneracy of the bound states. The symmetry category and category transition are observable from the photocurrent spectroscopy or optical spectrum. For simplicity's sake, in this paper, we only focus on inter-subband transition spectra, but the methodology can be extended to the inter-band transition cases. We predict that from the spectral measurements, the quantum dots' geometric information may be inversely extracted.

In the context of bacteria and models of their evolution under genome rearrangement, we explore a novel application of group representation theory to the inference of evolutionary history. Our contribution is to show, in a very general maximum likelihood setting, how to use elementary matrix algebra to sidestep intractable combinatorial computations and convert the problem into one of eigenvalue estimation amenable to standard numerical approximation techniques., 13 pages

We have used the SmallGroups library of groups, together with the computer algebra systems GAP and Mathematica, to search for groups with a three-dimensional irreducible representation in which one of the group generators has a twice-degenerate eigenvalue while another generator has non-degenerate eigenvalues. By assuming one of these group generators to commute with the charged-lepton mass matrix and the other one to commute with the neutrino (Dirac) mass matrix, one derives group-theoretical predictions for the moduli of the matrix elements of either a row or a column of the lepton mixing matrix. Our search has produced several realistic predictions for either the second row, or the third row, or for any of the columns of that matrix., 23 pages, 1 figure, added references, new appendices concerning the nilpotent groups and groups with a normal Sylow 3-subgroups, conclusions unchanged; some minor changes, matches published version

The appearance of linear spaces, describing physical quantities by vectors and tensors, is ubiquitous in all of physics, from classical mechanics to the modern notion of local Lorentz invariance. However, as natural as this seems to the physicist, most computer scientists would argue that something like a "local linear tangent space" is not very typical and in fact a quite surprising property of any conceivable world or algorithm. In this paper, we take the perspective of the computer scientist seriously, and ask whether there could be any inherently information-theoretic reason to expect this notion of linearity to appear in physics. We give a series of simple arguments, spanning quantum information theory, group representation theory, and renormalization in quantum gravity, that supports a surprising thesis: namely, that the local linearity of space-time might ultimately be a consequence of the linearity of probabilities. While our arguments involve a fair amount of speculation, they have the virtue of being independent of any detailed assumptions on quantum gravity, and they are in harmony with several independent recent ideas on emergent space-time in high-energy physics., 16 pages, 2 figures. Invited contribution to JPA's special issue "emerging talents". v2: close to published version

We derive, using the quaternionic group representation theory, the possible forms of the t-violating Hamiltonians, and consider time evolution in a two-level system when a t-violating term is added to a t-preserving unperturbed Hamiltonian. Transition probabilities are calculated; moreover, the time evolution of a complex state is determined analytically, showing that a quaternionic component arises naturally in its evolution. Finally, some properties of the time-reversal observable are studied and a set of selection rules is obtained.

A general definition for a symmetry group of an imaging system is given. A key requirement is that the operators that represent the symmetries in data space are conformal. The result is that the space of consistency conditions is invariant under the action of the given symmetry group. Via the theory of group representations, this fact provides information about the possible forms that these consistency conditions can take. The theory is illustrated by example for the 2D and 3D Radon transforms, the cone-beam transform on a circular orbit and the 2D attenuated Radon transform.

The time development of harmonic oscillator Wigner functions under quadratic Hamiltonians, including Markoffian damping terms, for thermal and squeezed reservoirs, is analysed by the use of group-theoretical methods.

A set of operations is found which permit the construction of a series of equations in finite group with the property that the functions which count their solutions are characters or generalized characters of the group.

Transformation matrices of phonon and phason strains under symmetry groups of two-dimensional (2D) quasicrystals (QCs) which are three-dimensional solids periodically stacked by aperiodic planes have been derived by using group representation theory. Quadratic invariants have been calculated for all 2D QCs of rank 5 and rank 7.

For the Ising model on the Cayley tree, we find new weakly periodic Gibbs measures corresponding to normal subgroups of index two in the group representation of the Cayley tree of order k ≤ 5.

The authors recently derived a family of nonlinear Schrodinger equations on R3 from fundamental considerations of generalized symmetry: ih(cross) delta t psi =-(h(cross)2/2m) Del 2 psi +F( psi , psi ) psi +ih(cross)D( Del 2 psi +( mod Del psi mod 2/ mod psi mod 2) psi ), where F is an arbitrary real functional and D a real, continuous quantum number. These equations, descriptive of a quantum mechanical current that includes a diffusive term, correspond to unitary representations of the group Diff(M) parametrized by D, where M=R3 is the physical space. In the present paper we explore the most natural ansatz for F, which is labelled by five real coefficients. We discuss the invariance properties, describe the stationary states and some non-stationary solutions, and determine the extra, dissipative terms that occur in the Ehrenfest theorem. We identify an interesting, Galilean-invariant subfamily whose properties we investigate, including the case where the dissipative terms vanish.

The general phenomenological free energy is derived for a liquid crystalline elastomer under arbitrary strain and orientational distortions. Using the group representations method we obtained all invariants, describing the coupling of translational and orientational deformations and/or external electric field. It is shown that in centrosymmetric materials (nematic rubbers) the only contribution to the free energy which is linear in (small) gradients of the director, is the coupling with electric field and the strain tensor (16 independent terms analogous to the flexoelectric effect). In chiral materials (cholesterics) the electric field couples with the strain tensor for uniform nematic director (piezoelectric effect, 3 invariants) and, in the absence of an electric field, translational and linear orientational distortions interact with each other (couple-stress effects, 6 invariants). Experimental observations and previous models are critically analysed.

The following groups are considered: the automorphism group of a Lebesgue measure space (with finite or -finite measure), groups of measurable functions with values in a Lie group, and diffeomorphism groups of manifolds. It turns out that the theory of representations of all these groups is closely related to the theory of representations of some category, which will be called "the category of -polymorphisms". Objects of this category are measure spaces, and a morphism from to is a probability measure on , where is a fixed Lie group. For some of the above-mentioned infinite-dimensional groups it is shown that any representation of extends canonically to a representation of some category of -polymorphisms. For automorphism groups of measure spaces this makes it possible to obtain a classification of all unitary representations. Also "new" examples of representations of groups of area-preserving diffeomorphisms of two-dimensional manifolds are constructed.

It is proved that the braid group representation constructed in terms of the quantum group can be covered by that obtained from the proposal of weight conservation by the extended Kauffman diagram technique. The non-standard braid group representations associated with SU(2) are obtained by adding the terms of weight conservation to the standard universal R-matrix.

When constructing polynomial bases for group representations, one is led naturally to consider a 'stretched product' of states. Elementary multiplets are states that cannot be expressed as the stretched product of any other two states, and they can be identified with terms occurring in certain generating functions. In this paper the authors define an analogous stretched product of Young tableaux together with elementary tableaux. These may be used in a manner similar to elementary multiplets to construct generating functions. They consider several examples where this method can be applied and in particular give new large-N (rank-free) branching rule generating functions for the group-subgroup pairs SU(n)*SU(m) contained in/implied by SU(n+m) (first four labels non-zero), SO(n) contained in/implied by SU(n) (first three labels non-zero) and Sp(2n) contained in/implied by SU(2n) (first five labels non-zero). This method of finding these generating functions uses the theory of Grobner bases which allows the authors to make precise the notion of compatibility relations or forbidden products of elementary tableaux.

The flux-breaking mechanism as applied to spacetimes of the form Rm*Td is re-examined in order to ascertain the effects of introducing a non-zero temperature into the model. By expressing the 1-loop effective potential in terms of Bessel functions and performing a high-temperature expansion, it is found that the entire fermion contribution and the non-leading terms in the expansion of the Yang-Mills contribution are exponentially damped as T to infinity . Consequently, the dominant term in the high-temperature effective potential on Rm*Td*T is found to behave in the manner of a pure Yang-Mills model on Rm*Td and hence the gauge symmetry remains unbroken. At low temperature, the effective potential behaves as per the zero temperature model on Rm+R1*Td hence, as outlined in a previous paper, the gauge symmetry can be broken by fermions in a suitable group representation.

A group representation generating the Mackey action is considered. Its decomposition into irreducible representations is studied; this leads to the classical Riesz products and their generalizations (also noncommutative). Some connections with the tail σ-algebra of random walks on the groups are indicated. Bibliography: 12 titles.

The paper provides an introduction into p-mechanics, which is a consistent physical theory suitable for a simultaneous description of classical and quantum mechanics. p-Mechanics naturally provides a common ground for several different approaches to quantisation (geometric, Weyl, coherent states, Berezin, deformation, Moyal, etc.) and has a potential for expansions into field and string theories. The backbone of p-mechanics is solely the representation theory of the Heisenberg group. Keywords: Classical mechanics, quantum mechanics, Moyal brackets, Poisson brackets, commutator, Heisenberg group, orbit method, deformation quantisation, symplectic group, representation theory, metaplectic representation, Berezin quantisation, Weyl quantisation, Segal--Bargmann--Fock space, coherent states, wavelet transform, contextual interpretation, string theory, field theory., Comment: LaTeX2e (amsart), 22 pages, 4 PS illustrations, v2 & v3 contain minor changes

A quantum particle in a impenetrable cubic well potential presents accidental degeneracy when the group is considered to be the symmetry group of the system. This degeneracy becomes natural when a new symmetry group, embedding the group, is proposed. This new group turns out to be the semidirect product , where is a two-dimensional compact continuous group whose generators correspond to linear combinations of the one-dimensional Hamiltonians. The systematic degeneracy is studied in detail, the new group is identified and its irreducible representations (irreps) are constructed by means of induction, an approach that allows the irreducibility and completeness to be assured. Pythagorean degeneracy as well as the one due to commensurable sides are not considered.

The group of symplectic linear canonical transformations has an integral kernel which has quadratic and linear phases, and which is realized by the geometric paraxial optical model. The discrete counterpart of this model is a finite Hamiltonian system that acts on N-point signals through N × N matrices whose elements also have a constant absolute value, although they do not form a representation of that group. Those matrices that are also unitary are Hadamard matrices. We investigate the manifolds of these N × N matrices under the equivalence imposed by the model, and find them to be on two-sided cosets. By means of an algorithm we determine representatives that lead to collections of mutually unbiased bases.

We use the SmallGroups Library to find the finite subgroups of U(3) of order smaller than 512 which possess a faithful three-dimensional irreducible representation. From the resulting list of groups we extract those groups that can not be written as direct products with cyclic groups. These groups are the basic building blocks for models based on finite subgroups of U(3). All resulting finite subgroups of SU(3) can be identified using the well known list of finite subgroups of SU(3) derived by Miller, Blichfeldt and Dickson at the beginning of the 20th century. Furthermore we prove a theorem which allows to construct infinite series of finite subgroups of U(3) from a special type of finite subgroups of U(3). This theorem is used to construct some new series of finite subgroups of U(3). The first members of these series can be found in the derived list of finite subgroups of U(3) of order smaller than 512. In the last part of this work we analyse some interesting finite subgroups of U(3), especially the group S_4(2)\cong A_4\rtimes Z_4, which is closely related to the important SU(3)-subgroup S_4., 36 pages, 1 figure; errors corrected; some comments and references added

In this paper we determine the fusion rules of the logarithmic ${\calW}_{p,q}$ triplet theory and construct the Grothendieck group with subgroups for which consistent product structures can be defined. The fusion rules are then used to determine projective covers. This allows us also to write down a candidate for a modular invariant partition function. Our results demonstrate that recent work on the ${\cal W}_{2,3}$ model generalises naturally to arbitrary (p,q)., 24 pages; v2: Minor expansion of the "Note Added". Version to appear in J. Phys. A

This paper considers a class of inverse problems involving deconvolution density estimation over the Euclidean motion group. Group representations of the Euclidean motion group are used to break apart convolution products, followed by compression and Tikhonov regularization to invert the distortion operator which is assumed known and compact. The integrated mean-squared error for the deconvolution density estimator is calculated whereby polynomial bounds on the recovery are obtained.

Pairs (G, Γ) are considered, where G is a vector space over an arbitrary fixed field and Γ is a group for which there is defined a representation with respect to G. A class of such pairs is called a prevariety if it is saturated and closed under the operations of forming subpairs and Cartesian products. A prevariety is called bounded if the variety generated by it differs from the class of all pairs. A prevariety is called small if it is generated by a single pair. The following theorems are proved. Theorem 1. Over any field the semigroup of bounded prevarieties of pairs is free. Theorem 2. The small prevarieties of pairs are indecomposable and generate a free semigroup of prevarieties.

Group cohomology techniques are used to derive conditions for a group G to be a local splitting group for G. As a first step the exactness of the inflation-restriction sequence gives a characterisation of factor systems arising in locally operating representations of transitive transformation groups. Some examples and applications of the theory are also given.

The operators r2, p2, and r.p+p.r are a basis of a representation of the algebra O2,1. Using harmonic oscillator wavefunctions with a fixed angular momentum as basis of the representation space, the isometrically equivalent representation in work by Barut (1967) can be identified. This allows one to write down formulae for matrix elements, relative to oscillator wavefunctions, of any operator of the corresponding group representation. These include exp(icr2), exp(iq2p2/h(cross)2), their products, and the scaling operator. One can thus obtain an algebraic derivation of the matrix elements of a momentum-dependent nuclear interaction proposed by Tabakin and Davies (1966).

In order to unify in a single invariant group of the Quantum Electrodynamics the Poincare group and the gauge group, we study some type of topological extension of the Poincare group by the gauge group, seen as the additive group of an Hilbert space where is realized the representation of mass zero and spin zero of the Poincare group. It is shown that the semi-direct product is the only possible extension.

We study the properties of spinors in n dimensions and the consequences of taking Dirac matrices to be Grassmann valued. We find that this yields representations of the sympletic group in n dimensions, which can be interpreted as representations of the orthogonal group in - n dimensions. In particular we find the sympletic analogues of spinorial representations. We also prove that the relation Sp (n) SO (-n) holds in general.

Overlap integrals for vibronic transitions with doubly and triply degenerate vibrations are calculated in closed form by means of the coherent state method. New recurrence relations are obtained for the overlap integrals. The possibility of regarding the overlap integral as a matrix element of some operator of a dynamical group representation leads to new sum rules for the Franck-Condon factors.

The Hamiltonian H of a quantum system of n particles is considered in spaces of functions that are transformed by multiple irreducible representations of a symmetry group of H, namely the direct product of the symmetric group by an arbitrary compact subgroup of the full rotation group. Sufficient conditions are found for the discrete spectrum of H to be infinite. The results obtained permit one in many cases to reduce this problem to that for an operator of a two-particle system. Bibliography: 18 titles.

A group representation is a pair (A,Γ), where A is a module over a commutative ring K with identity and Γ is a group that acts on A. In the category of group representations over a fixed K, both A and Γ are variable. We study varieties in this category. This paper is a survey of results and problems in this area and connections with other topics.

A so-called stochastic method is introduced which can be used to decompose the reducible representations of any finite group into irreducible representations. In this way, the regular representation of a finite group is reduced completely to obtain all of the irreducible representations of the group. It also induces a new approach to calculate the character table of a finite group. Based on this method, these calculations or decompositions can be performed by means of a computer.

The method of an earlier paper (see ibid., vol.20, p.4075 (1987)) is applied to the non-compact case to solve a family of second Poschl-Teller Morse-Rosen and Eckart equations with quantised coupling constants. Both discrete and continuous spectra, bound state and scattering wavefunctions (transmission coefficients) are found from the matrix elements of group representations,.

The problem of decomposing the symmetrized powers of representations of point groups is discussed in detail. Advantage is taken of the fact that each point group representation is closely related either to an induced linear character or to a representation of SU(2). The theory introduced by Gard in 1973 is developed to cover the symmetrization of induced linear characters and also representations induced from a normal subgroup.

This survey is timed to coincide with the sixtieth birthday of I. M. Gel'fand. The authors have confined themselves to those branches of mathematics in which he has been engaged during the last decade, and in the various branches the chronology of the articles covered by the survey is different. Gel'fand's research in the theory of group representations, which has lasted for thirty years, falls into several cycles; the majority of his results are widely known and were dealt with in the survey [1*] on the occasion of his fiftieth birthday. For this reason we deal here only with the results of the last ten years. His first articles on integral geometry appeared more than ten years ago, but this branch of mathematics is still in a formative phase. Because of this we include in the survey an outline of Gel'fand's basic research in integral geometry, not excluding some that is comparatively old. Topology is a new branch of his scientific activity; all the work in this field was carried out between 1968 and 1973. In the preparation of the survey we had the help of I. N. Bernshtein, M. I. Graev, and D. A. Kazhdan.

This article contains the evaluation of the image of the natural homomorphism from the even-dimensional Wall group into the ring of complex representations of a finite group . The computations are carried out for finite groups acting freely and linearly on spheres, by means of a differential-topological interpretation of ; the Atiyah-Singer invariant is utilized as a tool.Bibliography: 8 titles.

It is shown that, in addition to the number of distinct equivalence classes, the representation characters of a 'labelling' can also be derived using standard combinatorial methods. These characters provide additional information on the structure of the equivalence classes which is relevant in physical applications.

In this article, we investigate a special class of representations of discrete groups. Making use of a number of geometrical constructions, we establish various relations between the Wall group of the fundamental group of a manifold, the higher signatures, and the K-theory of the classifying space of the fundamental group.