Point Group Tables

Point Group Tables

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Point Group Tables of T(23)

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Character Table of the group T(23)*
T(23)	#	1	2	3⁺	3^-	functions
Mult.	-	1	3	4	4	·
A	Γ₁	1	1	1	1	x²+y²+z²
¹E ²E	Γ₂ Γ₃	1 1	1 1	w w²	w² w	(2z²-x²-y²,x²-y²)
T	Γ₄	3	-1	0	0	(x,y,z),(xy,xz,yz),(J_x,J_y,J_z)

w = exp(2iπ/3)

**Subgroups of the group T(23)**
Subgroup	Order	Index
T(23)	12	1
C₃(3)	3	4
D₂(222)	4	3
C₂(2)	2	6
C₁(1)	1	12

[ Subduction tables ]

**Multiplication Table of irreducible representations of the group T(23)**
T(23)	A	¹E	²E	T
A	A	¹E	²E	T
¹E	·	²E	A	T
²E	·	·	¹E	T
T	·	·	·	A+¹E+²E+2T

[ Note: the table is symmetric ]

**Symmetrized Products of Irreps**
T(23)	A	¹E	²E	T
[A x A]	1	·	·	·
[¹E x ¹E]	·	·	1	·
[²E x ²E]	·	1	·	·
[T x T]	1	1	1	1

**Antisymmetrized Products of Irreps**
T(23)	A	¹E	²E	T
{A x A}	·	·	·	·
{¹E x ¹E}	·	·	·	·
{²E x ²E}	·	·	·	·
{T x T}	·	·	·	1

**Irreps Decompositions**
T(23)	A	¹E	²E	T
V	·	·	·	1
[V²]	1	1	1	1
[V³]	1	·	·	3
[V⁴]	2	2	2	3
A	·	·	·	1
[A²]	1	1	1	1
[A³]	1	·	·	3
[A⁴]	2	2	2	3
[V²]xV	1	1	1	5
[[V²]²]	3	3	3	4
{V²}	·	·	·	1
{A²}	·	·	·	1
{[V²]²}	1	1	1	4

V ≡ the vector representation
A ≡ the axial representation

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

[ Note: x means allowed ]

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

[ Note: x means allowed ]

**Subduction of the rotation group D(L) to irreps of the group T(23)**
Irreps	Dimensions	Irreps of the point group
L	2L+1	A	¹E	²E	T
0	1	1	·	·	·
1	3	·	·	·	1
2	5	·	1	1	1
3	7	1	·	·	2
4	9	1	1	1	2
5	11	·	1	1	3
6	13	2	1	1	3
7	15	1	1	1	4
8	17	1	2	2	4
9	19	2	1	1	5
10	21	2	2	2	5

^* 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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