Matrix Elements Of Representations Research Articles

Upper estimates of the Morse numbers for the matrix elements of real linear irreducible representations of compact connected simple Lie groups

The Morse numbers of spaces of matrix elements for real irreducible linear representations of compact connected simple Lie groups are estimate from above in a variety of ways, in terms of the dimension, the Dynkin index of the representation, the eigenvalues of the invariant Laplace operator, and the volume of the group.

Симетрія коливальних станів та електронних π-орбі-талей молекули бензену C6H6. Тонка структура спінза-лежних розщеплень

Iз застосуванням оператора проектування на елементи матриць незвiдних представлень точкової групи симетрiї 6/mmm (D6ℎ) молекули бензену C6H6 побудовано аналiтичнi вирази та векторнi зображення форм всiх її нормальних коливань, у тому числi двократно вироджених. Знайдено характери представлень, що вiдповiдають симетрiї п-електронних орбiталей молекули бензену без урахування спiну електрона та проективних представлень її спiнорних п′-орбiталей. Представлення спiнорних п′-орбiталей молекули бензену C6H6 належать проективному класу K1 та описують тонку структуру вперше встановлених спiнзалежних при врахуваннi спiну електрона розщеплень станiв, вироджених без врахування спiну п-орбiталей.

Entropic inequalities for matrix elements of rotation group irreducible representations

Using the entropic inequalities for Shannon and Tsallis entropies new inequalities for some classical polynomials are obtained. To this end, an invertible mapping for the irreducible unitary representation of groups SU(2) and SU(1, 1) like Jacoby polynomials and Gauss’ hypergeometric functions, respectively, are used.

English

This paper is devoted to the study of matrix elements of irreducible representations of the enveloping deformed Heisenberg algebra with reflection, motivated by recurrence relations satisfied by hypergeometric functions. It is shown that the matrix elements of a suitable operator given as a product of exponential functions are expressed in terms of d-orthogonal polynomials, which are reduced to the orthogonal Meixner polynomials when d = 1. The underlying algebraic framework allowed a systematic derivation of the recurrence relations, difference equation, lowering and rising operators and generating functions which these polynomials satisfy.

Integral and series representations of special functions related to the group SO(2, 2)

In this sequel to our earlier works [3, 14, 15], we aim to present certain integral and series representations for special functions by using some different group theoretical methods as follows: Restrictions of the representation matrix elements to some block-diagonal matrices; Poisson transform intertwining two realizations of the SO(2, 2)-representation; Invariant properties of the bilinear integral functionals which are used to obtain the matrix elements of bases transforms operators.

Multi-Photon Multi-Channel Interferometry for Quantum Information Processing

This thesis reports advances in the theory of design, characterization and simulation of multi-photon multi-channel interferometers. I advance the design of interferometers through an algorithm to realize an arbitrary discrete unitary transformation on the combined spatial and internal degrees of freedom of light. This procedure effects an arbitrary $n_{s}n_{p}\times n_{s}n_{p}$ unitary matrix on the state of light in $n_{s}$ spatial and $n_{p}$ internal modes. I devise an accurate and precise procedure for characterizing any multi-port linear optical interferometer using one- and two-photon interference. Accuracy is achieved by estimating and correcting systematic errors that arise due to spatiotemporal and polarization mode mismatch. Enhanced accuracy and precision are attained by fitting experimental coincidence data to a curve simulated using measured source spectra. The efficacy of our characterization procedure is verified by numerical simulations. I develop group-theoretic methods for the analysis and simulation of linear interferometers. I devise a graph-theoretic algorithm to construct the boson realizations of the canonical SU$(n)$ basis states, which reduce the canonical subgroup chain, for arbitrary $n$. The boson realizations are employed to construct $\mathcal{D}$-functions, which are the matrix elements of arbitrary irreducible representations, of SU$(n)$ in the canonical basis. I show that immanants of principal submatrices of a unitary matrix $T$ are a sum of the diagonal $\mathcal{D}(\Omega)$-functions of group element $\Omega$ over $t$ determined by the choice of submatrix and over the irrep $(\lambda)$ determined by the immanant under consideration. The algorithm for $\mathrm{SU}(n)$ $\mathcal{D}$-function computation and the results connecting these functions with immanants open the possibility of group-theoretic analysis and simulation of linear optics.

Wilson’s type Hilbert space valued functional equations

AbstractIn the present paper we characterize, in terms of characters, additive functions and matrix elements of irreducible two-dimensional representations, the continuous solutions of some trigonometric functional equations for mappings defined on an arbitrary group and taking their values in a complex Hilbert space with the Hadamard product. In addition, we investigate a superstability result for these equations.

Algorithms for SU(n) boson realizations and D-functions

Boson realizations map operators and states of groups to transformations and states of bosonic systems. We devise a graph-theoretic algorithm to construct the boson realizations of the canonical SU(n) basis states, which reduce the canonical subgroup chain, for arbitrary n. The boson realizations are employed to construct D-functions, which are the matrix elements of arbitrary irreducible representations, of SU(n) in the canonical basis. We demonstrate that our D-function algorithm offers significant advantage over the two competing procedures, namely, factorization and exponentiation.

An Elementary Derivation of the Matrix Elements of Real Irreducible Representations of so(3)

Using elementary techniques, an algorithmic procedure to construct skew-symmetric matrices realizing the real irreducible representations of so(3) is developed. We further give a simple criterion that enables one to deduce the decomposition of an arbitrary real representation R of so(3) into real irreducible components from the characteristic polynomial of an arbitrary representation matrix.

Form factors in the algebraic cluster model

I present a derivation of form factors in the algebraic cluster model for an arbitrary number of identical clusters. The form factors correspond to representation matrix elements which are derived in closed form for the harmonic oscillator and deformed oscillator limits. These results are relevant for applications in nuclear, molecular and hadronic physics.

Characteristic identities for Lie (super)algebras

We present an overview of characteristic identities for Lie algebras and superalgebras. We outline methods that employ these characteristic identities to deduce matrix elements of finite dimensional representations. To demonstrate the theory, we look at the examples of the general linear Lie algebras and Lie superalgebras.

The multivariate Meixner polynomials as matrix elements of SO(d, 1) representations on oscillator states

The multivariate Meixner polynomials are shown to arise as matrix elements of unitary representations of the SO(d, 1) group on oscillator states. These polynomials depend on d discrete variables and are orthogonal with respect to the negative multinomial distribution. The emphasis is put on the bivariate case for which the SO(2, 1) connection is used to derive the main properties of the polynomials: orthogonality relation, raising/lowering relations, generating function, recurrence relations and difference equations as well as explicit expressions in terms of standard (univariate) Krawtchouk and Meixner polynomials. It is explained how these results generalize directly to d variables.

The multivariate Krawtchouk polynomials as matrix elements of the rotation group representations on oscillator states

An algebraic interpretation of the bivariate Krawtchouk polynomials is provided in the framework of the three-dimensional isotropic harmonic oscillator model. These polynomials in two discrete variables are shown to arise as matrix elements of unitary reducible representations of the rotation group in three dimensions. Many of their properties are derived by exploiting the group-theoretic setting. The bivariate Tratnik polynomials of Krawtchouk type are seen to be special cases of the general polynomials that correspond to particular rotations involving only two parameters. It is explained how the approach generalizes naturally to (d + 1) dimensions and allows us to interpret multivariate Krawtchouk polynomials as matrix elements of SO(d + 1) unitary representations. Indications are given on the connection with other algebraic models for these polynomials.

Representation Theory over Tropical Semifield and Langlands Duality

Recently we propose a class of infinite-dimensional integral representations of classical \({\mathfrak{gl}_{\ell+1}}\)-Whittaker functions and local Archimedean local L-factors using two-dimensional topological field theory framework. The local Archimedean Langlands duality was identified in this setting with the mirror symmetry of the underlying topological field theories. In this note we introduce elementary analogs of the Whittaker functions and the Archimedean L-factors given by Ul+1-equivariant symplectic volumes of appropriate Kahler Ul+1-manifolds. We demonstrate that the functions thus defined have a dual description as matrix elements of representations of monoids \({GL_{\ell+1}(\mathcal{R}),\; \mathcal{R}}\) being the tropical semifield. We also show that the elementary Whittaker functions can be obtained from the non-Archimedean Whittaker functions over \({\mathbb{Q}_p}\) by taking the formal limit p→ 1. Hence the elementary special functions constructed in this way might be considered as functions over the mysterious field \({\mathbb{Q}_1}\). The existence of two representations for the elementary Whittaker functions, one as an equivariant volume and the other as a matrix element, should be considered as a manifestation of a hypothetical elementary analog of the local Langlands duality for number fields. We would like to note that the elementary local L-factors coincide with L-factors introduced previously by Kurokawa.

Permutational quantum computing

In topological quantum computation the geometric details of a particle trajectory are irrelevant; only the topology matters. Taking this one step further, we consider a model of computation that disregards even the topology of the particle trajectory, and computes by permuting particles. Whereas topological quantum computation requires anyons, permutational quantum computation can be performed with ordinary spin-1/2 particles, using a variant of the spin-network scheme of Marzuoli and Rasetti. We do not know whether permutational computation is universal. It may represent a new complexity class within BQP. Nevertheless, permutational quantum computers can in polynomial time approximate matrix elements of certain irreducible representations of the symmetric group and approximate certain transition amplitudes from the Ponzano-Regge spin foam model of quantum gravity. No polynomial time classical algorithms for these problems are known.

Calculation of Metric for Gaussian Weave State

Using the recoupling theorem and graph calculation in loop quantum gravity, it is demonstrated that the action of metric matrix operator on Gaussian weave state is an eigenaction, the representation matrix elements of the metric operator and their expectation values are calculated. The values of length of tangent vectors of edges adjacent to the vertex of Gaussian weave state, as well as the angles between them are also obtained in the cases of k=0 and k=2.

Eigenaction of metric operator on Gaussian weave state and spin-geometry

In the recouping theorem and the graph calculation for loop quantum gravity, it is proved that the action of metric matrix operator on Gaussian weave state is an eigenaction, and the representation matrix elements of the metric operator and their expectation values are calculated. The values of the length of tangent vectors with 4 edges (P=1) adjacent to the vertex of Gaussian weave state ψp, as well as the angles between them, are also obtained in the cases of k=0 and k=2.

Relativistic spherical functions on the Lorentz group

Matrix elements of irreducible representations of the Lorentz group are calculated on the basis of complex angular momentum. It is shown that Laplace-Beltrami operators, defined in this basis, give rise to Fuchsian differential equations. An explicit form of the matrix elements of the Lorentz group has been found via the addition theorem for generalized spherical functions. Different expressions of the matrix elements are given in terms of hypergeometric functions both for finite-dimensional and unitary representations of the principal and supplementary series of the Lorentz group.

The Hook Fusion Procedure

We derive a new expression for the diagonal matrix elements of irreducible representations of the symmetric group. We obtain this new expression using Cherednik's fusion procedure. However, instead of splitting Young diagrams into their rows or columns, we consider their principal hooks. This minimises the number of auxiliary parameters needed in the fusion procedure.

General Solutions of Relativistic Wave Equations

General solutions of relativistic wave equations are studied in terms of functions on the Lorentz group. A close relationship between hyperspherical functions and matrix elements of irreducible representations of the Lorentz group is established. A generalization of the Gel'fand-Yaglom theory for higher-spin equations is given. A two-dimensional complex sphere is associated with each point of Minkowski spacetime. The separation of variables in a general relativistically invariant system is obtained via the hyperspherical functions defined on the surface of the two-dimensional complex sphere. In virtue of this the wave functions are represented in the form of series in hyperspherical functions. Such a description allows one to consider all the physical fields on an equal footing. General solutions of the Dirac and Weyl equations, and also the Maxwell equations in the Majorana-Oppenheimer form, are given in terms of functions on the Lorentz group.