Complete description of the Voronoï cell of the Lie Algebras An weight lattice. On the Bounds for the number of d-faces of the n-dimensional Voronoï cells
ALGEBRES DE LIE
POLYGONES DE VORONOI
Version étendue d'une conférence faite le 9 janvier 1997 au Centre de Recherches Mathématiques de l'Université de Montréal au cours du colloque pour le 60e anniversaire de Jiri Patera et Pavel Winternitz.
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.
MICHEL
P/97/53
IHES
07/1997
A4
11 f.
EN
TEXTE
PREPUBLICATION
P_97_53.pdf
1997
Physical implications of crystal symmetry and time reversal
RESEAUX CRISTALLINS
SYMETRIE
ESPACE ET TEMPS
Lectures given at the international school on Symmetry and Structural Properties of Condensed Matter. August 28 - September 5, 1996, Zajaczkowo (Posnan), Poland.
MICHEL
P/96/80
IHES
12/1996
A4
13 f.
EN
TEXTE
PREPUBLICATION
P_96_80.pdf
1996
Bravais classes, Voronoï cells, Delone symbols
CRISTALLOGRAPHIE
MATHEMATIQUES
GROUPES SPATIAUX
POLYGONES DE VORONOI
CLASSES DE BRAVAIS
SYMBOLES DE DELONE
This text correspond to a set of lectures given at Third international school at Zajaceskowo (Poznan) Polane on Symmetry and structural Properties of condensed matter
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.
MICHEL
P/94/63
IHES
12/1994
A4
21 f.
EN
TEXTE
PREPUBLICATION
P_94_63.pdf
1994
Chirurgie des grassmanniennes
VARIETES DE GRASSMANN
COMPACTIFICATIONS
PAVAGE
MATHEMATIQUES
VARIETES DE SCHUBERT
ESPACES FIBRES
LAFFORGUE
M/02/31
IHES
05/2002
A4
134 f.
FR
TEXTE
PREPUBLICATION
M_02_31.pdf
2002