To Arkady Vainshtein on his 60th anniversary

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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]]>

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.]]>

Abstract :Denoting these bounds by Nd(n), 0 ? d ? n we prove that Nd(n)/(n + 1)! is a polynomial Pd(n) of degree d with rational coefficients. We give the polynomials for d ? 5 explicitly. The proof uses the fact that these bounds Nd(n) are also the number of d-faces of the Voronoï cell of the weight lattice of the Lie algebra An (it is also the Cayley diagram of the symmetric group Sn+1, which is isomorphic to the Weyl group of An). Each d-face of this cell is a zonotope that can be defined by a symmetry group ~ G d (?), (d-dimensional reflection subgroup of the A n Weyl group. We show that for a given d and n large enough, all such subgroups of A n are represented, and we compute explicitly N(G d (?),n) the number of d-faces of type G d (?) in the Voronoï cell of L = A w n. The final result is obtained by summing over a. That also yields the simple expression Nd(n)=(n+1?d)!S(n+1?d)n+1Nd(n)=(n+1?d)!Sn+1(n+1?d) where the last symbol is the Stirling number of second kind.]]>