Point Group Tables

Point Group Tables

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Point Group Tables of C_i(-1)

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Character Table of the group C_i(-1)*
C_i(-1)	#	1	-1	functions
A_g	Γ₁⁺	1	1	x²,y²,z²,xy,xz,yz,J_x,J_y,J_z
A_u	Γ₁^-	1	-1	x,y,z

**Subgroups of the group C_i(-1)**
Subgroup	Order	Index
C_i(-1)	2	1
C₁(1)	1	2

[ Subduction tables ]

**Multiplication Table of irreducible representations of the group C_i(-1)**
C_i(-1)	A_g	A_u
A_g	A_g	A_u
A_u	·	A_g

[ Note: the table is symmetric ]

**Symmetrized Products of Irreps**
C_i(-1)	A_g	A_u
[A_g x A_g]	1	·
[A_u x A_u]	1	·

**Antisymmetrized Products of Irreps**
C_i(-1)	A_g	A_u
{A_g x A_g}	·	·
{A_u x A_u}	·	·

**Irreps Decompositions**
C_i(-1)	A_g	A_u
V	·	3
[V²]	6	·
[V³]	·	10
[V⁴]	15	·
A	3	·
[A²]	6	·
[A³]	10	·
[A⁴]	15	·
[V²]xV	·	18
[[V²]²]	21	·
{V²}	3	·
{A²}	3	·
{[V²]²}	15	·

V ≡ the vector representation
A ≡ the axial representation

**IR Selection Rules**
IR	A_g	A_u
A_g	·	x
A_u	x	·

[ Note: x means allowed ]

**Raman Selection Rules**
Raman	A_g	A_u
A_g	x	·
A_u	·	x

[ Note: x means allowed ]

**Subduction of the rotation group D(L) to irreps of the group C_i(-1)**
Irreps	Dimensions	Irreps of the point group
L	2L+1	A_g	A_u
0	1	1	·
1	3	·	3
2	5	5	·
3	7	·	7
4	9	9	·
5	11	·	11
6	13	13	·
7	15	·	15
8	17	17	·
9	19	·	19
10	21	21	·

^* 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 Ci(-1)

Point Group Tables of C_i(-1)