Irreducible Tensor Representations Research Articles

Ternary algebras associated with irreducible tensor representations of SO(3) and the quark model

We show that each irreducible tensor representation of weight 2 of the rotation group in the space of rank 3 tensors over three-dimensional space gives rise to an associative algebra with unity. We find the algebraic conditions to be satisfied by the generators of these algebras. Part of these relations is of binary, and another part is of ternary type. The structure of the latter one is based on the use of the cyclic group [Formula: see text] generated by the primitive cubic root of unity given by [Formula: see text]. The subspace of each algebra spanned by triple products of generators is five-dimensional (5D) and is identical with the space of irreducible tensor representation of weight 2 of the rotation group [Formula: see text]. We define a Hermitian scalar product in this 5D subspace and construct its orthonormal basis in terms of triple products of generators. Then we find an explicit formula for the Lie algebra homomorphism [Formula: see text]. We suggest that the algebras constructed in this way, with binary and ternary constitutive relations, may find applications in the quark model and the Grand Unified Theories.

Convexity criteria for expansions of the Landau-de Gennes elastic free energy

The Landau-de Gennes elastic free energy of liquid crystals is a functional expressed in terms of a symmetric traceless tensor order parameter. We derive convexity criteria for SO(3)-invariant expansions of the elastic energy that are quadratic in the gradient and up to fourth-order overall in the tensor order parameter. To do so we write the integrand as the sum of squares of linearly independent spherical tensors from which definiteness criteria can be ascertained. In particular, transformations between Cartesian and spherical tensors representations are computed based on irreducible tensor representations of the rotation group.

Neutron spin dynamics in polarized targets

We present neutron elastic scattering amplitude for arbitrary polarized target in irreducible spherical tensor representation. The general approach for the description of neutron spin dynamics for the propagation trough the medium with an arbitrary polarization is discussed in a relation to the search for time reversal invariance violation in neutron scattering.

First fullyab initiopotential energy surface of methane with a spectroscopic accuracy

Full 9-dimensional ab initio potential energy surfaces for the methane molecule are constructed using extended electronic structure coupled-cluster calculations with various series of basis sets following increasing X cardinal numbers: cc-pVXZ (X = 3, 4, 5, 6), aug-cc-ACVXZ (X = 3, 4, 5), and cc-pCVXZ-F12 (X = 3, 4). High-order dynamic electron correlations including triple and quadrupole excitations as well as relativistic and diagonal Born-Oppenheimer breakdown corrections were accounted for. Analytical potential functions are parametrized as non-polynomial expansions in internal coordinates in irreducible tensor representation. Vibrational energy levels are reported using global variational nuclear motion calculations with exact kinetic energy operator and a full account of the tetrahedral symmetry of CH4. Our best ab initio surface including above-mentioned contributions provides the rms (obs.-calc.) errors of less than 0.11 cm−1 for vibrational band centers below 4700 cm−1, and ∼0.3 cm−1 for all 229 assigned experimentally determined vibrational levels up to the Icosad range <7900 cm−1 without empirically adjusted parameters. These results improve the accuracy of ab initio methane vibrational predictions by more than an order of magnitude with respect to previous works. This is an unprecedented accuracy of first-principles calculations of a five-atomic molecule for such a large data set. New ab initio potential results in significantly better band center predictions even in comparison with best available empirically corrected potential energy surfaces. The issues related to the basis set extrapolation and an additivity of various corrections at this level of accuracy are discussed.

S-functions, spectral functions of hyperbolic geometry, and vertex operators with applications to structure for Weyl and orthogonal group invariants

In this paper we analyze the quantum homological invariants (the Poincaré polynomials of the slN link homology). In the case when the dimensions of homologies of appropriate topological spaces are precisely known, the procedure of the calculation of the Kovanov–Rozansky type homology, based on the Euler–Poincaré formula can be appreciably simplified. We express the formal character of the irreducible tensor representation of the classical groups in terms of the symmetric and spectral functions of hyperbolic geometry. On the basis of Labastida–Mariño–Ooguri–Vafa conjecture, we derive a representation of the Chern–Simons partition function in the form of an infinite product in terms of the Ruelle spectral functions (the cases of a knot, unknot, and links have been considered). We also derive an infinite-product formula for the orthogonal Chern–Simons partition functions and analyze the singularities and the symmetry properties of the infinite-product structures.

Gel’fand–Zetlin basis for a class of representations of the Lie superalgebra

A new, so called odd Gel’fand–Zetlin (GZ) basis is introduced for the irreducible covariant tensor representations of the Lie superalgebra . The related GZ patterns are based upon the decomposition according to a particular chain of subalgebras of . This chain contains only genuine Lie superalgebras of type with k and l nonzero (apart from the final element of the chain which is ). Explicit expressions for a set of generators of the algebra on this GZ basis are determined. The results are extended to an explicit construction of a class of irreducible highest weight modules of the general linear Lie superalgebra .

First Predictions of Rotationally Resolved Infrared Spectra of Dideuteromethane (12CH2D2) From Potential Energy and Dipole Moment Surfaces

We report the variationally computed infrared spectrum of (12)CH2D2 using our recent potential energy and dipole moment methane surfaces, which have been initially derived in the irreducible tensor representation adapted to the tetrahedral symmetry of the major isotopologue (12)CH4. The nuclear motion calculations are accomplished by combining the normal-mode Eckart-Watson Hamiltonian with isotopic and symmetry transformations. Our direct vibrational calculations are compared to the 93 observed band centers up to 6300 cm(-1). Except for two outliers the root-mean-square deviation is 0.22 cm(-1) and the maximum error is 0.7 cm(-1) without empirical adjustment of parameters. The work aims at filling the gap concerning missing line strength information for this molecule. Theoretical spectra predictions are given up to J = 25 and, for the very first time, ab initio intensity predictions for rovibrational line transitions are in good qualitative agreement with available experimental spectra.

Extensions of inhomogeneous polynomial representations for $\mathfrak {sl}(m+1|n)$sl(m+1|n)

We extend an inhomogeneous polynomial representation of special linear Lie superalgebras $\mathfrak {sl}(m+1|n)$sl(m+1|n) to an inhomogeneous representation on the tensor space of any irreducible tensor representation of the general linear Lie superalgebras $\mathfrak {gl}(m|n)$gl(m|n) with this polynomial space via Shen's mixed product. We find a sufficient and necessary condition for irreducibility of these inhomogeneous representations, and get the Jordan-H$\ddot{\mbox{o}}$ölder series of those which are not irreducible. Furthermore, we point out that all these irreducible modules are lowest weight modules. Their lowest weights and character formulas are obtained.

The Hopf algebra structure of the character rings of classical groups

The character ring of covariant irreducible tensor representations of the general linear group admits a Hopf algebra structure isomorphic to the Hopf algebra of symmetric functions. Here we study the character rings and of the orthogonal and symplectic subgroups of the general linear group within the same framework of symmetric functions. We show that and also admit natural Hopf algebra structures that are isomorphic to that of , and hence to . The isomorphisms are determined explicitly, along with the specification of standard bases for and analogous to those used for . A major structural change arising from the adoption of these bases is the introduction of new orthogonal and symplectic Schur–Hall scalar products. Significantly, the adjoint with respect to multiplication no longer coincides, as it does in the case, with a Foulkes derivative or skew operation. The adjoint and Foulkes derivative now require separate definitions, and their properties are explored here in the orthogonal and symplectic cases. Moreover, the Hopf algebras and are not self-dual. The dual Hopf algebras and are identified. Finally, the Hopf algebra of the universal rational character ring of mixed irreducible tensor representations of the general linear group is introduced and its structure maps identified.

Rotational and vibrational energy levels of methyl fluoride calculated from a new potential energy surface

A new potential energy surface of methyl fluoride is constructed using extended ab initio CCSD(T) calculations with the cc-pVQZ basis at 5100 nuclear configurations. Its analytical representation is determined through an expansion in symmetry adapted products of orthogonal coordinates involving 600 parameters up to 6th order. A good convergence for variational calculations of vibrational levels of the CH3F molecule was obtained with a RMS(obs.-calc.) deviation of less than 4cm−1 for fundamental band centers. The equilibrium geometry of the ab initio PES was empirically optimized using experimental J=1 energy levels for four isotopologues 12CH3F, 13CH3F, 12CD3F, and 13CD3F. The resulting variational calculations with the full normal mode Hamiltonian in the irreducible tensor representation gave a RMS(obs.-calc) deviation of 0.00036cm−1 for rotational energies up to J=5 for the major isotopologue. This represents a considerable improvement with respect to available global predictions of vibration and rotational levels of methyl fluoride.

Gel’fand–Zetlin basis and Clebsch–Gordan coefficients for covariant representations of the Lie superalgebra gl(m∣n)

A Gel’fand–Zetlin basis is introduced for the irreducible covariant tensor representations of the Lie superalgebra gl(m∣n). Explicit expressions for the generators of the Lie superalgebra acting on this basis are determined. Furthermore, Clebsch–Gordan coefficients corresponding to the tensor product of any covariant tensor representation of gl(m∣n) with the natural representation V([1,0,…,0]) of gl(m∣n) with highest weight (1,0,…,0) are computed. Both results are steps for the explicit construction of the parastatistics Fock space.

Irreducible tensor representations of general linear Lie superalgebras

We present a description of irreducible tensor representations of general linear Lie superalgebras in terms of generalized determinants in the symmetric and exterior superalgebras of a superspace over a field of characteristic zero.

A BGG-Type Resolution for Tensor Modules over General Linear Superalgebra

We construct a Bernstein–Gelfand–Gelfand type resolution in terms of direct sums of Kac modules for the finite-dimensional irreducible tensor representations of the general linear superalgebra. As a consequence it follows that the unique maximal submodule of a corresponding reducible Kac module is generated by its proper singular vector.

Collective multipolelike signatures of entanglement in symmetricN-qubit systems

A cogent theory of collective multipole-like quantum correlations in symmetric multiqubit states is presented by employing SO(3) irreducible spherical tensor representation. An arbitrary bipartite division of this system leads to a family of inequalities to detect entanglement involving averages of these tensors expressed in terms of the total system angular momentum operator. Implications of this theory to the quantum nature of multipole-like correlations of all orders in the Dicke states are deduced. A selected set of examples illustrate these collective tests. Such tests detect entanglement in macroscopic atomic ensembles, where individual atoms are not accessible.

Analogue of Kostant's u-cohomology formula for the general linear superalgebra

We determine in an explicit form the Lie superalgebra cohomology groups of the upper triangular odd subalgebra (Glm|n)+1 of the general linear superalgebra Glm|n with coefficients in irreducible tensor modules over Glm|n. Using this, we develop a Weyl-type character formula for the irreducible tensor representations. Both in the computation of the cohomology groups and in the construction of the character formula, the Weyl group of the infinite-rank affine Kac-Moody algebra of A-type plays an essential role.

Gelfand–Tsetlin Pattern and Strict Partitions

We give a representation-theoretical interpretation of the Gelfand–Tsetlin pattern for strict partitions. Using the Howe duality involving a pair of the queer Lie superalgebras and an analogue of the Littlewood–Richardson rule for Schur Q-functions, we show that such patterns give the branching rule for the irreducible tensor representations of the queer Lie superalgebra.

Vector (e, e′γ)correlations in ionization–excitation of He by electron impact

The general description of simultaneous electron-impact ionization–excitationprocesses is reformulated in terms of an irreducible tensor representation. Using anumerical model, in which the interaction between a ‘fast’ ionizing projectile andthe target is described by first-or second-order plane-wave and distorted-wavemodels, combined with a convergent close-coupling-type description of the initialbound state and the interaction between a ‘slow’ ejected electron andthe residual ion, new benchmark calculations for the (e, e′γ)process have been performed. The predictions are compared with recentexperimental data on ionization–excitation of helium. The similarity to results forexcitation processes and the potential extension of propensity rules is discussed.

Collisional decoherence in the presence of ultrafast optical pulses

An expression for the van der Waals interaction between an active atom and a perturber atom is derived using an irreducible tensor representation. This interaction leads to magnetic state decoherence in an ensemble of active atoms. The magnetic state decoherence can be suppressed by the application of ultrafast optical pulses. It is shown that the depolarization cross section can be suppressed by as much as 40% with only four optical pulses during a single collision and that, for a large number of pulses, the cross section is proportional to ${n}_{0}^{\ensuremath{-}2/11}$ where ${n}_{0}$ is the number of pulses at the Weisskopf radius. This paper generalizes our previous results in which the collisional interaction was modeled as a square pulse.

The Irreducible Tensor Representations of gl(m|1) and Their Generic Homology

The Irreducible Tensor Representations of gl(m|1) and Their Generic Homology

Nonlinear optical properties of polyenoctupoles: a multipolar tensorial quantum analysis

This paper presents a theoretical analysis of the first-order hyperpolarizability properties ( β) of polyenic molecules of octupolar C 3h symmetry (polyenoctupoles), based on CNDO/S calculations, and interpreted in the frame of a tensorial formalism. One of the goals of this analysis is to dissect the hyperpolarizability components in such a way as to reveal interactions within the different subunits constituting the octupolar system. To this end, the polyenoctupole properties are compared to those of dipolar analogues having the same conjugation length m. The general laws obtained for 1D dipolar polyenes (linear variations of λ with m , exponential variations of β with m) are also verified for the polyenoctupoles. The theory also reveals that, for the same conjugation length, the polyenoctupoles should exhibit a better hyperpolarizability–absorption trade-off than the corresponding 1D polyenes. A multipolar tensorial analysis of the dipolar and octupolar components of β in 1D polyenes demonstrates that the nonlinear anisotropy ρ of the 1D polyenes effectively corresponds to ideal 1D chromophores, with a octupolar/dipolar ratio close to its theoretical value of 2/3 =0.82. Finally, the octupolar interaction concept is introduced within the frame of irreducible tensor representation. In this generally applicable representation, the β tensor of a multipolar system is dissected into two separate tensor components, one ( β A) representing the sum of the contributions of the individual dipole subunits, the other ( β I) reflecting interactions (charge transfer or Coulombic) between the dipole units constituting the multipole. The interaction term β I could be determined in magnitude and in direction in the irreducible J=3 space. In the C 3h octupoles considered in the present work, the octupolar interaction term β I is found negligible for the smallest molecules and becomes relatively important (up to 20–30% of β A) only for the octupoles having the largest size.