International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 1.2, p. 69

Table 1.2.7.2 

M. Ephraïm,b T. Janssen,a A. Jannerc and A. Thiersd

Table 1.2.7.2| top | pdf |
Calculation with characters

GeneratorComposite characterCharactersDecomposition
[\pmatrix{ 0&1&0\cr 0&0&1\cr 1&0&0 }] R E A AA    
[\chi (R)] 3 0 0    
[\chi (R)^3] 27 0 0    
[\chi (R^2)] 3 0 0    
[\chi (R^2)\chi (R)] 9 0 0    
[\chi (R^3)] 3 3 3    
Example (1) [{{1}\over{6}}(\chi (R)^3 +3\chi (R^2)\chi (R)+2\chi (R^3))] 10 1 1   [4D_1+3D_2+3D_3 ]
[\pmatrix{ 0&-1 \cr 1& -1 } ] R E A AA    
[\chi (R)] 2 −1 −1    
[\chi (R)^2] 4 1 1    
[\chi (R^2)] 2 −1 −1    
Example (2) [{{1}\over{2}} (\chi (R)^2 +\chi (R^2)]) 3 0 0   [D_1+D_2+D_3]
[\pmatrix{ 0&-1&0\cr 1&0&0 \cr 0&0&1 }] R E A AA AAA  
[\chi (R)] 3 1 −1 1  
[\chi (R)^2] 9 1 1 1  
[\chi (R^2)] 3 −1 3 −1  
Example (3) [{{1}\over{2}} (\chi (R)^2 +\chi (R^2)]) 6 0 2 0 [2D_1+D_2+2D_3+D_4 ]
As above [\chi (R)] 3 1 −1 1  
Example (4) [\chi (R)^3] 27 1 −1 1  
  [\chi (R^2)] 3 −1 3 −1  
  [\chi (R^2)\chi (R)] 9 −1 −3 −1  
  [\chi (R^3)] 3 1 −1 1  
  [{{1}\over{6}}(\chi (R)^3 +3\chi (R^2)\chi (R)+2\chi (R^3))] 10 0 −2 0 [2D_1+3D_2+2D_3+3D_4]
As above [\chi (R)] 3 1 −1 1  
Example (5) [{{1}\over{2}} (\chi (R)^2 +\chi (R^2)])=[\chi_s (R)] 6 0 2 0  
  [\chi_{s}(R)^2] 36 0 4 0  
  [\chi_{s}(R^2)] 6 2 6 2  
  ((12)(34)) 21 1 5 1 [7D_1+4D_2+6D_3+4D_4]
As above, example (6) [{{1}\over{2}} (\chi (R)^2 -\chi (R^2)]) 3 1 −1 1 [D_1+D_2+D_4 ]
As above, example (7) [{{1}\over{6}}(\chi (R)^3 -3\chi (R^2)\chi (R)+2\chi (R^3))] 1 1 1 1 [D_1]