Point Group Tables

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Point Group Tables of S₄(-4)

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Character Table of the group S₄(-4)*
S₄(-4)	#	1	2	-4⁺	-4^-	functions
A	Γ₁	1	1	1	1	x²+y²,z²,J_z
B	Γ₂	1	1	-1	-1	z,xy,x²-y²
¹E ²E	Γ₄ Γ₃	1 1	-1 -1	-1j 1j	1j -1j	(x,y),(xz,yz),(J_x,J_y)

**Subgroups of the group S₄(-4)**
Subgroup	Order	Index
S₄(-4)	4	1
C₂(2)	2	2
C₁(1)	1	4

[ Subduction tables ]

**Multiplication Table of irreducible representations of the group S₄(-4)**
S₄(-4)	A	B	¹E	²E
A	A	B	¹E	²E
B	·	A	²E	¹E
¹E	·	·	B	A
²E	·	·	·	B

[ Note: the table is symmetric ]

**Symmetrized Products of Irreps**
S₄(-4)	A	B	¹E	²E
[A x A]	1	·	·	·
[B x B]	1	·	·	·
[¹E x ¹E]	·	1	·	·
[²E x ²E]	·	1	·	·

**Antisymmetrized Products of Irreps**
S₄(-4)	A	B	¹E	²E
{A x A}	·	·	·	·
{B x B}	·	·	·	·
{¹E x ¹E}	·	·	·	·
{²E x ²E}	·	·	·	·

**Irreps Decompositions**
S₄(-4)	A	B	¹E	²E
V	·	1	1	1
[V²]	2	2	1	1
[V³]	2	2	3	3
[V⁴]	5	4	3	3
A	1	·	1	1
[A²]	2	2	1	1
[A³]	2	2	3	3
[A⁴]	5	4	3	3
[V²]xV	4	4	5	5
[[V²]²]	7	6	4	4
{V²}	1	·	1	1
{A²}	1	·	1	1
{[V²]²}	3	4	4	4

V ≡ the vector representation
A ≡ the axial representation

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

[ Note: x means allowed ]

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

[ Note: x means allowed ]

**Subduction of the rotation group D(L) to irreps of the group S₄(-4)**
Irreps	Dimensions	Irreps of the point group
L	2L+1	A	B	¹E	²E
0	1	1	·	·	·
1	3	·	1	1	1
2	5	1	2	1	1
3	7	2	1	2	2
4	9	3	2	2	2
5	11	2	3	3	3
6	13	3	4	3	3
7	15	4	3	4	4
8	17	5	4	4	4
9	19	4	5	5	5
10	21	5	6	5	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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Point Group Tables of S4(-4)

Point Group Tables of S₄(-4)