$\mathsf{u(2) \supset su^*(2)\supset G}$ symmetry adaptation for powers of $\mathsf{E}$ irreducible representations of point groups

2004 ◽  
Vol 30 (2) ◽  
pp. 181-199 ◽  
Author(s):  
F. Michelot ◽  
M. Rey
Keyword(s):  
Irreducible Representations ◽  
Point Groups ◽  
Symmetry Adaptation

scholarly journals Emergence of colour symmetry in free-vibration acoustic resonance of a nonlinear hyperelastic material

2013 ◽  
Vol 469 (2159) ◽  
pp. 20130275 ◽  
Author(s):  
Ryuichi Tarumi
Keyword(s):  
Free Vibration ◽  
Ritz Method ◽  
Geometrical Nonlinearity ◽  
Acoustic Resonance ◽  
Finite Amplitude ◽  
Hyperelastic Material ◽  
Irreducible Representations ◽  
Hyperelastic Materials ◽  
Point Groups ◽  
Linearized System

We investigated free-vibration acoustic resonance (FVAR) of two-dimensional St Venant–Kirchhoff-type hyperelastic materials and revealed the existence and structure of colour symmetry embedded therein. The hyperelastic material is isotropic and frame indifferent and includes geometrical nonlinearity in its constitutive equation. The FVAR state is formulated using the principle of stationary action with a subsidiary condition. Numerical analysis based on the Ritz method revealed the existence of four types of nonlinear FVAR modes associated with the irreducible representations of a linearized system. Projection operation revealed that the FVAR modes can be classified on the basis of a single colour (black or white) and three types of bicolour (black and white) magnetic point groups: , , and . These results demonstrate that colour symmetry naturally arises in the finite amplitude nonlinear FVAR modes, and its vibrational symmetries are explained in terms of magnetic point groups rather than the irreducible representations that have been used for linearized systems. We also predicted a grey colour nonlinear FVAR mode which cannot be derived from a linearized system.

THE TETRAHEDRAL–OCTAHEDRAL BASES FOR THE GENERALIZED ROTOR

2011 ◽  
Vol 20 (02) ◽  
pp. 565-568 ◽  
Author(s):  
A. GÓŹDŹ ◽  
A. SZULERECKA ◽  
A. DOBROWOLSKI
Keyword(s):  
Irreducible Representations ◽  
Angular Momenta ◽  
Point Groups

The rotational basis for irreducible representations of the point groups T d, O and O h for nuclear angular momenta J = 0, 1…, 5 is constructed.

Allowable irreducible representations of the point groups with five-fold rotational axes

Pramana ◽  
2012 ◽  
Vol 79 (6) ◽  
pp. 1533-1541
Author(s):  
RAMA MOHANA RAO K ◽  
SIMHACHALAM B ◽  
HEMAGIRI RAO P
Keyword(s):  
Irreducible Representations ◽  
Point Groups

On the symmetries of spherical harmonics

1963 ◽  
Vol 255 (1054) ◽  
pp. 199-215 ◽  
Keyword(s):  
Irreducible Representation ◽  
Spherical Harmonics ◽  
Irreducible Representations ◽  
Linear Combinations ◽  
Point Groups

This paper extends the work described in a previous paper by one of the authors (Altmann 1957). The spherical harmonics that belong to the irreducible representations of the cubic groups are now given up to and including l = 12. Also, for all point groups the expansions in spherical harmonics that are given belong to the separate columns of the irreducible representations (whereas before they were linear combinations of such functions). Accordingly, full tables for the irreducible representations for all crystallographic point groups are required and are given in the paper. Finally, a technique is described, and used throughout in the tables, to orthogonalize several expansions that belong to the same column of the same irreducible representation. Therefore, the different expansions listed in the tables are always fully orthogonal.

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