Abstract : We discuss the general problem of symmetry beakdown in the algebraic approach to statistical mechanics. We consider in particular the case of classical quantum lattice systems.]]>
*) These notes are a result of discussions between S. Dpolicher, G. Gallavotti and D. Ruelle, they are not intended for publication.]]> 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.]]>
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]]>
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.]]> 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.]]>
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.]]> 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. ]]> E/H yield a classification of the topologically stable defects and configurations of these ordered media. This suggests a predictive value of this scheme for yet unobserved media and for defects.]]>
Abstract. - Topologically stable symmetry defects in ordered media can be classified by some homotopy groups. As an example we establish a complete classification of such defects for crystals.]]>