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

International Tables for Crystallography (2006). Vol. D, ch. 1.2, p. 57

Table 1.2.6.4 

T. Janssena*

aInstitute for Theoretical Physics, University of Nijmegen, 6524 ED Nijmegen, The Netherlands
Correspondence e-mail: ted@sci.kun.nl

Table 1.2.6.4| top | pdf |
Irreducible representations for dihedral groups [D_{n}]

(a) n odd. [m=1,\ldots, (n-1)/2;\,j=1,\ldots, (n-1)/2], s.c.m = smallest common multiple.

 [\varepsilon][\alpha^{j}][\ldots][\beta]
[n_i]112n
Order1s.c.m.([n,j])[\ldots]2
[ \Gamma_{1}] 1 1 [\ldots] 1
[\Gamma_{2}] 1 1 [\ldots] −1
[\Gamma_{2+m}] 2 [2\cos(2\pi mj/n) ] [\ldots] 0

(b) n even. [m=1,\ldots, (n/2- 1);\,j=1,\ldots, (n/2- 1)], s.c.m = smallest common multiple.

 [\varepsilon][\alpha^{n/2}][\alpha^{j}][\ldots][\beta][\alpha \beta]
[n_{i}]112[\ldots][n/2][n/2]
Order12s.c.m.[(n,j)][\ldots]22
[ \Gamma_{1}] 1 1 1 [\ldots] 1 1
[\Gamma_{2}] 1 1 1 [\ldots] −1 −1
[\Gamma_{3}] 1 [(-1)^{n/2}] [(-1)^{j}] [\ldots] 1 −1
[\Gamma_{4}] 1 [(-1)^{n/2}] [(-1)^{j}] [\ldots] −1 1
[\Gamma_{4+m}] 2 [(-1)^m]2 2cos([2 \pi mj/n]) [\ldots] 0 0