Ray representations of point groups and the irreducible representations of space groups and double space groups

1966 ◽  
Vol 260 (1108) ◽  
pp. 1-36 ◽  
Keyword(s):  
Finite Group ◽  
Irreducible Representations ◽  
High Symmetry ◽  
Full Matrix ◽  
Space Groups ◽  
Double Space ◽  
Point Groups ◽  
A Finite Group ◽  
Representations Of Space ◽  
Complex Cases

The theory of the ray representations of a finite group is summarized and full matrix ray representations are derived and tabulated for all thirty-two point groups. It is shown that any irreducible representation of any of the 230 space groups and of the corresponding double groups may be obtained quickly and easily from these ray representations of the point groups. The most complex cases which arise, namely points of high symmetry on the surface of the Brillouin zone for the regular holohedric space groups, 01 ... OJ°, are treated explicitly. The relation of the present work to the recent treatments of Slater and Kovalev is discussed.

Classes d'éléments conjugués des groupes cristallographiques

1978 ◽  
Vol 34 (6) ◽  
pp. 895-900
Author(s):  
J. Sivardière
Keyword(s):  
Finite Group ◽  
Double Point ◽  
Invariant Subgroup ◽  
Factor Group ◽  
Direct Products ◽  
Simple Point ◽  
Space Groups ◽  
Point Groups ◽  
A Finite Group

Let G be a finite group, H an invariant subgroup and F the corresponding factor group. The classes of conjugated elements of G are derived from the classes of H and F. We consider simple point groups and symmorphic space groups, which are semi-direct products H^F, then double point groups and non- symmorphic space groups, which are extensions of F by H.

L'application des representations au calcul explicite sur certaines chaînes de Markov finies

1984 ◽  
Vol 16 (3) ◽  
pp. 656-666 ◽  
Author(s):  
Bernard Ycart
Keyword(s):  
Finite Group ◽  
Random Walk ◽  
Irreducible Representations ◽  
Permutation Groups ◽  
A Finite Group

We give here concrete formulas relating the transition generatrix functions of any random walk on a finite group to the irreducible representations of this group. Some examples of such explicit calculations for the permutation groups A4, S4, and A5 are included.

scholarly journals A Krull-Schmdit type theorem for coherent sheaves

2014 ◽  
Vol 22 (2) ◽  
pp. 51-56
Author(s):  
A. S. Argáez
Keyword(s):  
Finite Group ◽  
Projective Variety ◽  
Type Theorem ◽  
Coherent Sheaf ◽  
Irreducible Representations ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Coherent Sheaves ◽  
A Finite Group ◽  
Natural Decomposition

AbstractLet X be projective variety over an algebraically closed field k and G be a finite group with g.c.d.(char(k), |G|) = 1. We prove that any representations of G on a coherent sheaf, ρ : G → End(ℰ), has a natural decomposition ℰ ≃ ⊕ V ⊗k ℱV, where G acts trivially on ℱV and the sum run over all irreducible representations of G over k.

scholarly journals Double crystallographic groups and their representations on the Bilbao Crystallographic Server

2017 ◽  
Vol 50 (5) ◽  
pp. 1457-1477 ◽  
Author(s):  
Luis Elcoro ◽  
Barry Bradlyn ◽  
Zhijun Wang ◽  
Maia G. Vergniory ◽  
Jennifer Cano ◽  
...  
Keyword(s):  
Irreducible Representations ◽  
Site Symmetry ◽  
Crystallographic Groups ◽  
Space Groups ◽  
Symmetry Approach ◽  
Wyckoff Position ◽  
Double Space ◽  
Elementary Band ◽  
Symmetry Relations ◽  
Symmetry Operations

A new section of databases and programs devoted to double crystallographic groups (point and space groups) has been implemented in the Bilbao Crystallographic Server (http://www.cryst.ehu.es). The double crystallographic groups are required in the study of physical systems whose Hamiltonian includes spin-dependent terms. In the symmetry analysis of such systems, instead of the irreducible representations of the space groups, it is necessary to consider the single- and double-valued irreducible representations of the double space groups. The new section includes databases of symmetry operations (DGENPOS) and of irreducible representations of the double (point and space) groups (REPRESENTATIONS DPGandREPRESENTATIONS DSG). The toolDCOMPRELprovides compatibility relations between the irreducible representations of double space groups at differentkvectors of the Brillouin zone when there is a group–subgroup relation between the corresponding little groups. The programDSITESYMimplements the so-called site-symmetry approach, which establishes symmetry relations between localized and extended crystal states, using representations of the double groups. As an application of this approach, the programBANDREPcalculates the band representations and the elementary band representations induced from any Wyckoff position of any of the 230 double space groups, giving information about the properties of these bands. Recently, the results ofBANDREPhave been extensively applied in the description of and the search for topological insulators.

scholarly journals On the Degrees of Irreducible Representations of a Finite Group

1951 ◽  
Vol 3 ◽  
pp. 5-6 ◽  
Author(s):  
Noboru Itô
Keyword(s):  
Finite Group ◽  
Irreducible Representation ◽  
Irreducible Representations ◽  
A Finite Group

In 1896 G. Frobenius proved: the degree of any (absolutely) irreducible representation of a finite group divides its order. This theorem was improved by I. Schur in 1904 as follows: the degree of any irreducible representation of a finite group divides the index of its centre.

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