Structure and classification of band representations

CRISTALLOGRAPHIE

SOLIDES

ASTRONOMIE

ORBITE

ETOILES

MODELISATION

ENERGIE

ATOMES

Abstract : Band representaitons in solids are investigated in the general framework of induced representations by using the concepts of orbits (stars) and strata (Wyckoff positions) in their construction and classification. The connection between band representations and irreducible representations of space groups is established by reducing the former in the basis of quasi-Bloch functions wich are eigenfunctions of translations but ar not, in general, eigenfunctions of the Hamiltonian. While irreducible representations of space group ar fnite-dimensional and are induced from infinite-order little groups Gk for vectors K in the Brillouin zone, band representations are infinite-dimensional adn are induced from finite-order little groups Gr for vectors r in the Wigner-Seitz cell. This connection between irreductible representations and band representations of space groups shedd new light on the duality properties of the Brillouin zone and the Wigner-Seitz cell. As an introduction to band representations the induced representations of point groups wich id applied to the investigation of th equivalency of band representations. Based on this connection and on the properties of the crystallographic point groups a necessary condition id established for the inequivalency of band representations induced from maximal isotropy groups. For using this condition there is need fro the induced representaitons of oint groups and a full list of them is given in the paper. One is especially interested in irreducible-band representations which form the elementary building bricks for band representations. From the point of view of the physics, irreducible-band representations correspond to energy bands with minimal numbers of branches. A method id developped for finding all the inequivalent irreducible-band representations of space groups by using the induction from maximal isotropy groups. As a rule the latter leads to inequivalent irreducible-band representations. There are, however, few exceptions to this rule. A full list of such exceptions is tabulated in the paper. With this list at hand one can construct all the different irreducible-band representations of 2-dimensional space groups. For them we list the continuity chords of all their irreducible-band representations.

MICHEL

BACRY

ZAK

P/86/35

IHES

06/1986

A4

46 f.

EN

TEXTE

PREPUBLICATION

P_86_35.pdf

1986

IHES

IHES

MICHEL

BACRY

ZAK

https://repo-archives.ihes.fr/FONDS_IHES/I_Prepublications/MICHEL/1964-1989/P_86_35/P_86_35.pdf

Oui

Bures-sur-Yvette

WYCKOFF

BLOCH

HAMILTON

BRILLOUIN

WIGNER

SEITZ

HILBERT

WEYL

HEISENBERG

EUCLIDE

CLOIZEAUX

FROBENIUS

MACKEY

POINCARE

LORENTZ

MINKOWSKI

DIRICHLET

FEDOROV

VORONOY

DELONE

BRAVAIS

ABEL

SYLOW

What is a Crystal ?

CRISTAUX

MATHEMATIQUES

PAVAGE

FONCTIONS

Abstract : An historical survey of our knowledge of crystal structure and of the mathematical problem of paving space with one or a few types of stones show that a crystal has not necessarily a translation lattice. The recently discovered crystals of alloys Al with 14% Mn have an ecosahedral symetry (incompatible qith translation lattice). Their electron micrographs show a structure similar to a Penrose tiling. It seems that long range order characterizing crystals can be obtained from projection of a thin slab of genuine crystal in higher dimension so that crystal physical properties can be described by quasi-periodic functions (as generalizing periodic ones) or may be almost periodic functions.

MICHEL

P/85/68

IHES

12/1985

A4

8 f.

EN

TEXTE

PREPUBLICATION

P_85_68.pdf

1985

IHES

IHES

MICHEL

https://repo-archives.ihes.fr/FONDS_IHES/I_Prepublications/MICHEL/1964-1989/P_85_68/P_85_68.pdf

Oui

Bures-sur-Yvette

STENO

HAU

WEISS

HESSEL

BRAVAIS

SCHONFLIES

FEDOROV

VON LAUE

DE BROGLIE

JANNER

SCHECHTMAN

BLECH

GRATIAS

CAHN

EULER

FOURIER

DIRICHLET

MINKOWSKI

DELONE

WIGENR

SETIZ

BRILLOUIN

WANG

KNUTH

ROBINSON

HAMMAN

DE BRUIGN

GRATIAS

PENROSE

MAC KAY

FRAMER

LEVINE

STEINHART

DUNEAU

KATZ

KUO

JENSEN

FIBONACCI

RUELLE

BOHR

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.]]>

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.

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

IHES

IHES

MICHEL

https://repo-archives.ihes.fr/FONDS_IHES/I_Prepublications/MICHEL/1990-1999/P_97_53/P_97_53.pdf

Oui

Bures-sur-Yvette

VORONOI

LIE

STIRLING

WEYL

FEDOROV

GRAM

EICLIDE

MINKOWSKI

DELONE

SELLING

CAYLEY

CARTAN

COXETER

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.]]>

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.

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

IHES

IHES

MICHEL

https://repo-archives.ihes.fr/FONDS_IHES/I_Prepublications/MICHEL/1990-1999/P_94_63/P_94_63.pdf

Oui

Bures-sur-Yvette

BRAVAIS

VORONOI

DELONE

FEDOROV

DIRICHLET

EULCIDE

KLEIN

MOZRZYMAS

SENECHAL

FRANKENHEIM

LAGRANGE

GAUSS

JACOBI

HERMITE

JORDAN

SCHONLFLIES

LORENTZ

SELLING

HESSEL

GALOIS