Abstract : We give the most refined intrinsic classification of three dimensional Euclidean lattices by combining the 14 Bravais classes, the 5 combinatorial types of Voronoï cells and the 24 Delone symbols. After recalling the fondamental concepts of group actions, we define Bravais classes and Voronoï celles in arbitrary dimension. We are quite explicit for the application to two and three dimensions.]]>
Résumé. — Quelques remarques non scientifiques mais importantes. Notion de brisure spontanée
de symétrie ; premier exemple reconnu (Jacobi 1834) et ses relations actuelles avec l'astronomie.
On donne deux critères servant à déterminer sur quels sous-groupes du groupe de symétrie peut se
briser spontanément une symétrie ; illustration dans les cristaux liquides et les particules élémentaires.
Pour ces derniers, après un survol des nombreuses découvertes des années récentes la théorie
de GUrsey est brièvement exposée.

Abstract. — After some non-scientific non-irrelevant remarks, the concept of broken symmetry is explained and illustrated by the first historical example (Jacobi ellipsoid 1834) still useful in astronomy. Two criteria are given for finding on which subgroups of the symmetry group a symmetry can be broken ; examples in liquid crystals and elementary particles. A brief survey of the recent progress in the latter domain and of the Gilrsey theory.]]> at the origin, and determine the little groups (= isotropy groups) of these minima ; they are the subgroups into which the symmetry can be broken in the transition. By an efficient strategy we reduce the study of the 153 equivalence classes of irreps to few cases (6). Moreover we do not need to study the minima of invariant polynomials,we simply apply Morse theory to find the possible little groups of minima. ]]> 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.]]>
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.]]>