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.]]>
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.]]>
Abstract : This paper studies the number of extrema (and their positions) of a countinuous Morse function on the Brillouin zone, when it is invariant by the point group symmetry of the crystal. Forty years ago, Vanhove had shown the importance of this problem in physics, but he could use only the crystal translational symmetry. In that case Morse theory predicts at least eight extrema. With the added use of general symmetry arguments we show that this number is larger for six of the 14 classes of Bravais lattices ; moreover it is possible to give the position of the extrema (and their nature) for 30 of the 73 arithmetic classes.This paper is written for a larger audience than that of solid state physicists ; it also defines carefully the necessary crystallographic concepts which are generally poorly understood in the solid state literature.]]>
Abstract : We present S. Lie's work on the symmetry of ODE (ordinary differential equations) : action of Diff 2(x,y), the Lie algebra of vector fields of the plane x, y, on the set of ODE ; general form of ODE with a given symmetry algebra ; computation of the symmetry algebra of a given equation. The original part of this conference studies the general linear equation of order n>2 (this has also be done independenty by Mohamed and Leach, ref. [9]). We also present in a more precise form the work of Lie on the finite dimensional subalgebras of Diff2. As an application, we classify the symmetries of second order equations.]]>