Table 1.2.7.2

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

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International Tables for Crystallography (2006). Vol. D. ch. 1.2, p. 69

M. Ephraïm,^b T. Janssen,^a A. Janner^c and A. Thiers^d

Table 1.2.7.2 | top | pdf |
Calculation with characters

Generator	Composite character	Characters
$[\pmatrix{ 0&1&0\cr 0&0&1\cr 1&0&0 }]$	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
$[\pmatrix{ 0&-1 \cr 1& -1 } ]$	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
$[\pmatrix{ 0&-1&0\cr 1&0&0 \cr 0&0&1 }]$	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
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
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
As above, example (6)	$[{{1}\over{2}} (\chi (R)^2 -\chi (R^2)]$ )	3	1	−1	1
As above, example (7)	$[{{1}\over{6}}(\chi (R)^3 -3\chi (R^2)\chi (R)+2\chi (R^3))]$	1	1	1	1