We develop exact expressions for the coefficients of series representations of translations and rotations of local and multipole fundamental solutions of the Helmholtz equation in spherical coordinates. These expressions are based on the derivation of recurrence relations, some of which, to our knowledge, are presented here for the first time. The symmetry and other properties of the coefficien...

The notion that elementary systems correspond to irreducible representations of the Poincaré group is the starting point for this paper, which then goes on to discuss how a semigroup for the time evolution of unstable states and resonances could emerge from the underlying Poincaré symmetry. Important tools in this analysis are the Clebsch-Gordan coefficients for the Poincaré group.

We derive necessary and sufficient conditions for an ambiskew polynomial ring to have a Hopf algebra structure of a certain type. This construction generalizes many known Hopf algebras, for example U(sl2), Uq(sl2) and the enveloping algebra of the 3-dimensional Heisenberg Lie algebra. In a torsion-free case we describe the finite-dimensional simple modules, in particular their dimensions and pr...

Computer codes are developed to calculate Clebsch–Gordan coefficients of SU(3) in both SU(2)and SO(3)-coupled bases. The efficiency of this code derives from the use of vector coherent state theory to evaluate the required coefficients directly without recursion relations. The approach extends to other compact semi-simple Lie groups. The codes are given in subroutine form so that users can inco...

We use homotopy theory to define certain rational coefficients characteristic numbers with integral values, depending on a given prime number q and positive integer t . We prove the first nontrivial degree formula and use it to show that existence of morphisms between algebraic varieties for which these numbers are not divisible by q give information on the degree of such morphisms or on zero c...

Representations of the quantum superalgebra U q [osp(1/2)] and their relations to the basic hypergeometric functions are investigated. We first establish Clebsch-Gordan decomposition for the superalgebra U q [osp(1/2)] in which the representations having no classical counterparts are incorporated. Formulae for these Clebsch-Gordan coefficients are derived, and it is observed that they may be ex...

Addition theorems for vector spherical harmonics require the determination of the Gaunt coefficients that appear in a linearization expansion of the product of two associated Legendre functions. This paper presents an algorithm for the efficient calculation of these coefficients through solving the most appropriate (lower triangular) linear system and derives all relevant recurrence relations n...

We present a supersymmetric model of fermion masses with SU(4)×SU(2)2×U(1)X gauge group with matter in fundamental and antisymmetric tensor representations only. The up, down, charged lepton and neutrino Yukawa matrices are distinguished by different Clebsch-Gordan coefficients due to contracting over SU(4) and SU(2)R indices. We obtain a hierarchical light neutrino mass spectrum with bi-large ...

Velleda Baldoni, Nicole Berline, Michèle Vergne. Local Euler-Maclaurin expansion of Barvinok valuations and Ehrhart coefficients of a rational polytope. Matthias Beck, Christian Haase, Bruce Reznick, Michèle Vergne, Volkmar Welker and Ruriko Yoshida. Integer Points in Polyhedra-Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics, American Mathematical Society, pp.1...

We propose an alternative for the Clebsch decomposition of currents in fluid mechanics, in terms of complex potentials taking values in a Kähler manifold. We reformulate classical relativistic fluid mechanics in terms of these complex potentials and rederive the existence of an infinite set of conserved currents. We perform a canonical analysis to find the explicit form of the algebra of conser...