The real symplectic groups in quantum mechanics and optics

We present a utilitarian review of the family of matrix groups Sp(2n, ℛ), in a form suited to various applications both in optics and quantum mechanics. We contrast these groups and their geometry with the much more familiar Euclidean and unitary geometries. Both the properties of finite group elements and of the Lie algebra are studied, and special attention is paid to the so-called unitary metaplectic representation of Sp(2n, ℛ). Global decomposition theorems, interesting subgroups and their generators are described. Turning ton-mode quantum systems, we define and study their variance matrices in general states, the implications of the Heisenberg uncertainty principles, and develop a U(n)-invariant squeezing criterion. The particular properties of Wigner distributions and Gaussian pure state wavefunctions under Sp(2n, ℛ) action are delineated.     

Nuclear spin species,spin statistical weights and rovibronic tunnelling splittings of non-rigid water hexamer

Powerful multinomial generating functions together with the character table of the six-dimensional hyperoctahedral wreath product group S₆[S₂] of the non-rigid water hexamer are employed to obtain the nuclear spin species, nuclear spin multiplets and the total nuclear spin statistical weights of the rovibronic-tunnelling levels for both deuterated and regular forms. The methods are composed of algebraic generating functions involving 531,441 nuclear spin functions for (D₂O)₆ in a group of 46,080 operations and 65 irreducible representations of the S₆[S₂] group. Although the deuterated form of the non-rigid water hexamer possesses nuclear spin populations in 58 of the 65 possible symmetries of tunnelling levels, for the regular water hexamer only 23 of the 65 symmetries are populated with nuclear spin functions. The tunnelling splitting correlations of rovibronic levels of the water hexamer have been obtained from the Saykally's semi-rigid (G₄) model to the fully non-rigid limit (G₄₆₀₈₀). The computed nuclear spin statistical weights for protonated form of (H₂O)₆ call for a reinterpretation of the previous assignment of the observed spectra. The results can also be applied to enumerate the hyperfine patterns using the nuclear spin multiplets and intensities in semi-rigid to fully non-rigid limits.     

Equivariant <Emphasis Type="Italic">K</Emphasis>-Theory,Generalized Symmetric Products,and Twisted Heisenberg Algebra

 For a space X acted on by a finite group Γ, the product space X ⁿ affords a natural action of the wreath product Γ _n =Γ ⁿ ⋊S _n . The direct sum of equivariant K-groups were shown earlier by the author to carry several interesting algebraic structures. In this paper we study the K-groups of Γ _n -equivariant Clifford supermodules on X ⁿ . We show that is a Hopf algebra and it is isomorphic to the Fock space of a twisted Heisenberg algebra. Twisted vertex operators make a natural appearance. The algebraic structures on ℱ⁻ _Γ(X), when Γ is trivial and X is a point, specialize to those on a ring of symmetric functions with the Schur Q-functions as a linear basis. As a by-product, we present a novel construction of K-theory operations using the spin representations of the hyperoctahedral groups. Received: 3 February 2001 / Accepted: 17 August 2002 Published online: 10 January 2003     

Theory of neutrino masses and mixing

We motivate the usage of finite groups as symmetries of the Lagrangian. After a presentation of basic group-theoretical concepts, we introduce the notion of characters and character tables in the context of irreducible representations and discuss their applications. We exemplify these theoretical concepts with the groups S ₄ and A ₄. Finally, we discuss the relation between tensor products of irreducible representations and Yukawa couplings and describe a model for tri-bimaximal lepton mixing based on A ₄.     

Solvation properties of Li+ and CI− in water: molecular dynamics simulation with a non-rigid model

Molecular dynamics (MD) simulations were carried out to investigate the solvation properties of Li⁺ and C1^? ions in water with a relatively accurate but rarely used non-rigid model, RWK2, in this study. A new set of ion-water interaction parameters was evaluated from the experimental data and first principles calculation results of stable clusters, Li⁺(H₂O) _n (n = 1- 6) and CI^?(H₂O) _n (n = 1–4). With the ion-water potential parameters evaluated from the data of the clusters and the water-water potential predetermined from the non-rigid RWK2 model, the structural (radial distribution functions, angular distribution functions, spatial distribution functions, coordination number), dynamical (residence time) and energetic properties of the ionic salvations in bulk water were studied through a comprehensive analysis of our MD simulation outputs. These results not only agree well with experimental data and first principles calculations, but also reveal some new insights into the microscopic ionic salvation processes.     

The Legendre Structure of the Parisi Formula

We study easy quantum groups, a combinatorial class of orthogonal quantum groups introduced by Banica–Speicher in 2009. We show that there is a countable descending chain of easy quantum groups interpolating between Bichon’s free wreath product with the permutation group S_n and a semi-direct product of a permutation action of S_n on a free product. This reveals a series of new commutation relations interpolating between a free product construction and the tensor product. Furthermore, we prove a dichotomy result saying that every hyperoctahedral easy quantum group is either part of our new interpolating series of quantum groups or belongs to a class of semi-direct product quantum groups recently studied by the authors. This completes the classification of easy quantum groups. We also study combinatorial and operator algebraic aspects of the new interpolating series.     

Random Unitary Matrices, Permutations and Painlevé

This paper is concerned with certain connections between the ensemble of n×n unitary matrices – specifically the characteristic function of the random variable tr(U) – and combinatorics – specifically Ulam's problem concerning the distribution of the length of the longest increasing subsequence in permutation groups – and the appearance of Painlevé functions in the answers to apparently unrelated questions. Among the results is a representation in terms of a Painlevé V function for the characteristic function of tr(U) and (using recent results of Baik, Deift and Johansson) an expression in terms of a Painlevé II function for the limiting distribution of the length of the longest increasing subsequence in the hyperoctahedral groups. Received: 2 December 1998 / Accepted: 12 May 1999     

A WEIL REPRESENTATION OF sp(4) REALIZED BY DIFFERENTIAL OPERATORS IN THE SPACE OF SMOOTH FUNCTIONS ON S2 × S1

In the space of complex-valued smooth functions on S² × S¹, we explicitly realize a Weil representation of the real Lie algebra sp(4) by means of differential generators. This representation is a rare example of highest weight irreducible representation of sp(4) all whose weight spaces are 1-dimensional. We also show how this space splits into the direct sum of irreducible sl(2)-submodules. Selected applications: complete classification of yrast-band energies in even-even nuclei, the dynamical symmetry in some collective models of nuclear structure, the mapping methods for simplifying initial problem Hamiltonians.     

Effects of fin pitch and tube diameter on the air-side performance of tube bank fin heat exchanger with the fins punched plane and curved rectangular vortex generators

Tube bank fin heat exchangers with vortex generators are widely used in the field of industrial applications. The effects of the fin pitch and the tube diameter on the air-side performance of the tube bank fin heat exchanger with plane rectangular vortex generators (PRVG) and curved rectangular vortex generators (CRVG) are experimentally studied in this paper. Performance comparison is carried out between the fins with PRVG and CRVG. The experimental results show that both PRVG and CRVG can effectively enhance heat transfer performance compared with the plain fin. Both the fin pitch and the tube diameter have obvious effect on f compared with the effect on Nu, especially for the fin with PRVG. The characteristics of Nu, f, and Nu/f^1/3 are different for the fins with PRVG and CRVG. The fin with CRVG has a better heat transfer performance than the fin with PRVG for all the cases studied in this paper.     

Spectral-Structural Correlations in the Complexes of Transitional Elements with an Incompletely Filled 5f Electron Shell and XO3 n−-Type Trigonal Oxoligands

The specifics of the kinematic and dynamic parameters of the XO₃ ^n– groups attached in a different way to a central atom have been determined by the methods of experimental and computational vibrational spectroscopy. The characteristic nature of individual vibrations has been revealed and recommendations for their use in a spectral-structural analysis have been given. Spectral criteria to estimate the degree of the dentate character of ligands in insular-type and polymeric structures are suggested.     

Non-Standard Schwinger Fermionic Representation of Unitary Group

The non-standard Schwinger fermionic representation of the unitary group is studied by using n-fermion operators. One finds that the Schwinger fermionic representation of the U(n) group is not unique when n≥3. In general, based on n-fermion operators, the non-standard Schwinger fermionic representation of the U(n) group can be established in a uniform approach, where all the generators commute with the total number operators. The Schwinger fermionic representation of U(C _n ^m ) group is also discussed.     

The Donald–Flanigan Problem for Finite Reflection Groups

The Donald–Flanigan problem for a finite group H and coefficient ring k asks for a deformation of the group algebra kH to a separable algebra. It is solved here for dihedral groups and Weyl groups of types B _n and D _n (whose rational group algebras are computed), leaving but six finite reflection groups with solutions unknown. We determine the structure of a wreath product of a group with a sum of central separable algebras and show that if there is a solution for H over k which is a sum of central separable algebras and if S _n is the symmetric group then (i) the problem is solvable also for the wreath product H S _n = H × ··· × H (n times) S _n and (ii) given a morphism from a finite Abelian or dihedral group G to S _n it is solvable also for H G. The theorems suggested by the Donald–Flanigan conjecture and subsequently proven follow, we also show, from a geometric conjecture which although weaker for groups applies to a broader class of algebras than group algebras.     

On the Noncommutative Geometry of Twisted Spheres

We describe noncommutative geometric aspects of twisted deformations, in particular of the spheres of Connes and Landi and of Connes and Dubois Violette, by using the differential and integral calculus on these spaces that is covariant under the action of their corresponding quantum symmetry groups. We start from multiparametric deformations of the orthogonal groups and related planes and spheres. We show that only in the twisted limit of these multiparametric deformations the covariant calculus on the plane gives, by a quotient procedure, a meaningful calculus on the sphere. In this calculus, the external algebra has the same dimension as the classical one. We develop the Haar functional on spheres and use it to define an integral of forms. In the twisted limit (differently from the general multiparametric case), the Haar functional is a trace and we thus obtain a cycle on the algebra. Moreover, we explicitly construct the *-Hodge operator on the space of forms on the plane and then by quotient on the sphere. We apply our results to even spheres and compute the Chern–Connes pairing between the character of this cycle, i.e. a cyclic 2n-cocycle, and the instanton projector defined in math.QA/0107070.     

Realisations of the representations of para-Fermi algebra in Fock space of Bose operators: part II

Using the method given in Part I of this series (this volume, pp. 109–114), Green's isomorphisms of the para-Fermi algebra F _2n ¹ with 2n generators into ∈ _2n+1 ^a(2) are constructed. All the representations of the para-Fermi algebra are realised in Fock space #x210B; _2n ¹ of 2ⁿ Bose opcrators.     

Hecke algebras at roots of unity and crystal bases of quantum affine algebras

We present a fast algorithm for computing the global crystal basis of the basic -module. This algorithm is based on combinatorial techniques which have been developed for dealing with modular representations of symmetric groups, and more generally with representations of Hecke algebras of typeA at roots of unity. We conjecture that, upon specializationq1, our algorithm computes the decomposition matrices of all Hecke algebras at an ^th root of 1.Partially supported by PRC Math-Info and EEC grant n⁰ ERBCHRXCT930400.     

Remarks on the Schur—Howe—Sergeev Duality

We establish a new Howe duality between a pair of two queer Lie superalgebras (q(m),q(n)). This gives a representation theoretic interpretation of a well-known combinatorial identity for Schur Q-functions. We further establish the equivalence between this new Howe duality and the Schur–Sergeev duality between q(n) and a central extension of the hyperoctahedral group H _k. We show that the zero-weight space of a q(n)-module with highest weight given by a strict partition of n is an irreducible module over the finite group parameterized by . We also discuss some consequences of this Howe duality.     

Paraboson coherent states

It is known that the defining relations of the orthosymplectic Lie superalgebra osp(1|2n) are equivalent to the defining (triple) relations of n pairs of paraboson operators b _i ^±. In particular, the “parabosons of order p” correspond to a unitary irreducible (infinite-dimensional) lowest weight representation V(p) of osp(1|2n). Recently we constructed these representations V(p) giving the explicit actions of the osp(1|2n) generators. We apply these results for the n = 2 case in order to obtain “coherent state” representations of the paraboson operators.     

Recoupled Polarization-Transfer Methods for Solid-State H–C Heteronuclear Correlation in the Limit of Fast MAS

An in-depth account of the effects of homonuclear couplings and multiple heteronuclear couplings is given for a recently published technique for ¹H–¹³C dipolar correlation in solids under very fast MAS, where the heteronuclear dipolar coupling is recoupled by means of REDOR π-pulse trains. The method bears similarities to well-known solution-state NMR techniques, which form the framework of a heteronuclear multiple-quantum experiment. The so-called recoupled polarization-transfer (REPT) technique is versatile in that rotor-synchronized ¹H–¹³C shift correlation spectra can be recorded. In addition, weak heteronuclear dipolar coupling constants can be extracted by means of spinning sideband analysis in the indirect dimension of the experiment. These sidebands are generated by rotor encoding of the reconversion Hamiltonian. We present generalized variants of the initially described heteronuclear multiple-quantum correlation (HMQC) experiment, which are better suited for certain applications. Using these techniques, measurements on model compounds with ¹³C in natural abundance, as well as simulations, confirm the very weak effect of ¹H–¹H homonuclear couplings on the spectra recorded with spinning frequencies of 25–30 kHz. The effect of remote heteronuclear couplings on the spinning-sideband patterns of CH_n groups is discussed, and ¹³C spectral editing of rigid organic solids is shown to be practicable with these techniques.     

Parton-hadron duality in event generators

The validity of local parton-hadron duality within the framework of HERWIG and JETSET event generators is investigated. We concentrate one ⁺ e ^– annihilations in LEP 2 energy range as these interactions provide theoretically the cleanest condition for the discussion of this concept. We conclude that the concept of local parton-hadron duality is not valid in either of the two generators considered.     

Influence of covalence and anion symmetry on the structure of small metal hydroxide clusters: Sodium versus silver hydroxide

An ab initio study of the Na_n(OH)_n, Na_n(OH)_n-1 ⁺, Ag_n(OH)_n, and Ag_n(OH)_n-1 ⁺ clusters with n up to four is presented. The results of this study show that, in accordance with experimental observations, the sodium hydroxide clusters are almost purely ionic, while the Ag-O bond exhibits a significant covalent character. The perturbation caused by the non-spherical OH^- group relatively to an atomic anion, as well as the influence on structures and energies of the covalent character of the metal-oxygen bond are determined. The appearance of metal-metal bonds in the silver hydroxide clusters is also discussed. Finally, the theoretical results obtained on the Na-OH clusters are compared to experimental results available on the dissociation of the Na_n(OH)_n-1 ⁺ clusters. Received 9 August 1999 and Received in final form 1st December 1999