Point Group Tables

Point Group Tables

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Point Group Tables of C_s(m)

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Character Table of the group C_s(m)*
C_s(m)	#	1	m	functions
A^'	Γ₁	1	1	x,y,x²,y²,z²,xy,J_z
A^''	Γ₂	1	-1	z,xz,yz,J_x,J_y

**Subgroups of the group C_s(m)**
Subgroup	Order	Index
C_s(m)	2	1
C₁(1)	1	2

[ Subduction tables ]

**Multiplication Table of irreducible representations of the group C_s(m)**
C_s(m)	A^'	A^''
A^'	A^'	A^''
A^''	·	A^'

[ Note: the table is symmetric ]

**Symmetrized Products of Irreps**
C_s(m)	A^'	A^''
[A^' x A^']	1	·
[A^'' x A^'']	1	·

**Antisymmetrized Products of Irreps**
C_s(m)	A^'	A^''
{A^' x A^'}	·	·
{A^'' x A^''}	·	·

**Irreps Decompositions**
C_s(m)	A^'	A^''
V	2	1
[V²]	4	2
[V³]	6	4
[V⁴]	9	6
A	1	2
[A²]	4	2
[A³]	4	6
[A⁴]	9	6
[V²]xV	10	8
[[V²]²]	13	8
{V²}	1	2
{A²}	1	2
{[V²]²}	7	8

V ≡ the vector representation
A ≡ the axial representation

**IR Selection Rules**
IR	A^'	A^''
A^'	x	x
A^''	x	x

[ Note: x means allowed ]

**Raman Selection Rules**
Raman	A^'	A^''
A^'	x	x
A^''	x	x

[ Note: x means allowed ]

**Subduction of the rotation group D(L) to irreps of the group C_s(m)**
Irreps	Dimensions	Irreps of the point group
L	2L+1	A^'	A^''
0	1	1	·
1	3	2	1
2	5	3	2
3	7	4	3
4	9	5	4
5	11	6	5
6	13	7	6
7	15	8	7
8	17	9	8
9	19	10	9
10	21	11	10

^* C. J. Bradley and A. P. Cracknell (1972) The Mathematical Theory of Symmetry in Solids Clarendon Press - Oxford
^* Simon L. Altmann and Peter Herzig (1994). Point-Group Theory Tables. Oxford Science Publications.

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Point Group Tables of Cs(m)

Point Group Tables of C_s(m)