Finally counter examples, with v arbitrarily large, are given to the Dzyaloshinskii conjecture that there exist no stable fixed points when the Landau potential depends on more than V = 3 parameters. ]]> L'utilisation du théorème de réciprocité de Frobenius simplifie l'étude de la décomposition de la représentation de vibration de Gk en somme directe de représentations irréductibles.]]>
Abstract : For systems with a symmetry group G, the description of physical phenomena corresponding to a representation of G, depends only on the image of this representation.The classification of the images of the unirreps (unitary irreductible representations) of the little space groups Gk is remarkably simple. The nearly four thousands inequivalent unirreps corresponding to high symmetry wave vectors k have only 37 inequivalent images.]]> are considered. For each point group a polynomial integrity basis for invariants and the basic polynomial vector fields are first given. Then, the strata are defined via equations and inequalities involving the integrity basis. Finally, equations for zeros of a covariant vector field are given on each stratum in terms of the integrity basis, which appears via coefficients in the expansion of the vector field on the vector-field basis. All the results are tabulated and an illustration using the cubic group is presented. Mathematical background sufficient for extensions of the results is also given.

Résumé. - Pour tous les groupes ponctuels (sous-groupes fermés) finis ou continus des groupes orthogonaux
0(2) et 0(3) nous donnons une base d’intégrité e?(x) pour les polynômes invariants et pour les champs de vecteurs polynomiaux, nous donnons les équations et inégalités définissant les strates (union des orbites de même type). Finalement, nous écrivons des équations donnant les zéros d’un champ de vecteurs covariants sur une strate donnée; ces équations sont linéaires dans les composantes (invariantes) du champ de vecteurs sur les e?(x), les coefficients de ces termes étant eux-mêmes des invariants. Tous ces résultats sont résumés dans des tables et illustrés par un exemple d’application physique de symétrie cubique. La méthode mathématique pour obtenir ces résultats est expliquée de façon à permettre au lecteur de l’appliquer à d’autres groupes.]]> of an n-component field (or order parameter} for n&4. Simple proofs of known results are given.
Among new results, we shou that when it exists the stable fixed point is unique; we give some precision
on its symmetry and on its attractor basin.]]>
Abstract : An energy band in a solid contains an infinite number of states which transform linearly as a space group representation induced from a finite dimensional representation of the isotropy group of a point in space. A band representation is elementary if it cannot be decomposed as a direct sum of band representations; it describes a single band. We give a complete classification of the inequivalent elementary band representations.]]>
Abstract : We introduce a new set of definitions of the fundamental concepts of n-dimensional crystallography. It is based on general properties of group actions. We also show how to determine the crystallographic system and the Bravais class of a cristallographic group.]]>
This paper will appear in part two of Symmetry in Nature, a volume in honour of Luigi Radicati di Brozolo, Sculoa Normale Superiore, Pisa, 1989]]> We calculate it now in the hope that it might be of help to the readers of the accompanying paper on SU(3)xSU(13) where the same formalism is used to discuss some physically interesting problems.
We believe that the geometrical approach presented in these papers will help clarifying the relations and the properties of the fundamental interactions.
It might be used for example to understand the geometrical basis of some recent attempts to compute Cabibbo's angle.]]> Last time I came to Sweden, cars were drivent on the left side of the roads. This week most cars I saw were driven on the right side. I am very grateful to the Nobel Foundation for this opportunity to observe this P violation. I may have found the cause of this P violation by looking at buses or trucks with the same inscription on both sides ; they are not symmetrically written.
I have also made a couter-experiment to verify this theory. In Tokyo I found an equal probability for cars to be driven on the left side or the right side of the streets and I did observe that identical texts were placed symmetrically on buses and trucks : the first character is near the front part on both sides so the same text must be read from left to right on one side and from right to left on the other side.
Since I have never been yet in a car undergoing a spontaneous CP transformation, I will be able to give you today my report on CP, T and CPT invariance.]]> To appear in Reviews of Modern Physics.]]>