Point Group Tables

Point Group Tables

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Point Group Tables of C₃(3)

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Character Table of the group C₃(3)*
C₃(3)	#	1	3⁺	3^-	functions
A	Γ₁	1	1	1	z,x²+y²,z²,J_z
¹E ²E	Γ₃ Γ₂	1 1	w² w	w w²	(x,y),(xz,yz),(x²-y²,xy),(J_x,J_y)

w = exp(2iπ/3)

**Subgroups of the group C₃(3)**
Subgroup	Order	Index
C₃(3)	3	1
C₁(1)	1	3

[ Subduction tables ]

**Multiplication Table of irreducible representations of the group C₃(3)**
C₃(3)	A	¹E	²E
A	A	¹E	²E
¹E	·	²E	A
²E	·	·	¹E

[ Note: the table is symmetric ]

**Symmetrized Products of Irreps**
C₃(3)	A	¹E	²E
[A x A]	1	·	·
[¹E x ¹E]	·	·	1
[²E x ²E]	·	1	·

**Antisymmetrized Products of Irreps**
C₃(3)	A	¹E	²E
{A x A}	·	·	·
{¹E x ¹E}	·	·	·
{²E x ²E}	·	·	·

**Irreps Decompositions**
C₃(3)	A	¹E	²E
V	1	1	1
[V²]	2	2	2
[V³]	4	3	3
[V⁴]	5	5	5
A	1	1	1
[A²]	2	2	2
[A³]	4	3	3
[A⁴]	5	5	5
[V²]xV	6	6	6
[[V²]²]	7	7	7
{V²}	1	1	1
{A²}	1	1	1
{[V²]²}	5	5	5

V ≡ the vector representation
A ≡ the axial representation

**IR Selection Rules**
IR	A	¹E	²E
A	x	x	x
¹E	x	x	x
²E	x	x	x

[ Note: x means allowed ]

**Raman Selection Rules**
Raman	A	¹E	²E
A	x	x	x
¹E	x	x	x
²E	x	x	x

[ Note: x means allowed ]

**Subduction of the rotation group D(L) to irreps of the group C₃(3)**
Irreps	Dimensions	Irreps of the point group
L	2L+1	A	¹E	²E
0	1	1	·	·
1	3	1	1	1
2	5	1	2	2
3	7	3	2	2
4	9	3	3	3
5	11	3	4	4
6	13	5	4	4
7	15	5	5	5
8	17	5	6	6
9	19	7	6	6
10	21	7	7	7

^* 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 C3(3)

Point Group Tables of C₃(3)