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Point Group Tables of S4(-4)

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Character Table of the group S4(-4) *
S4(-4)#12-4+-4-functions
AΓ11111x2+y2,z2,Jz
BΓ211-1-1z,xy,x2-y2
1E
2E
Γ4
Γ3
1
1
-1
-1
-1j
1j
1j
-1j
(x,y),(xz,yz),(Jx,Jy)



Subgroups of the group S4(-4)
SubgroupOrderIndex
S4(-4)41
C2(2)22
C1(1)14

[ Subduction tables ]

Multiplication Table of irreducible representations of the group S4(-4)
S4(-4)AB1E2E
AAB1E2E
BA2E1E
1EBA
2EB

[ Note: the table is symmetric ]


Symmetrized Products of Irreps
S4(-4)AB1E2E
[A x A]1···
[B x B]1···
[1E x 1E]·1··
[2E x 2E]·1··


Antisymmetrized Products of Irreps
S4(-4)AB1E2E
{A x A}····
{B x B}····
{1E x 1E}····
{2E x 2E}····


Irreps Decompositions
S4(-4)AB1E2E
V·111
[V2]2211
[V3]2233
[V4]5433
A1·11
[A2]2211
[A3]2233
[A4]5433
[V2]xV4455
[[V2]2]7644
{V2}1·11
{A2}1·11
{[V2]2}3444

V ≡ the vector representation
A ≡ the axial representation


IR Selection Rules
IRAB1E2E
Axxx
Bxxx
1Exxx
2Exxx

[ Note: x means allowed ]


Raman Selection Rules
RamanAB1E2E
Axxxx
Bxxxx
1Exxxx
2Exxxx

[ Note: x means allowed ]


Irreps Dimensions Irreps of the point group
Subduction of the rotation group D(L) to irreps of the group S4(-4)
L2L+1AB1E2E
011···
13·111
251211
372122
493222
5112333
6133433
7154344
8175444
9194555
10215655



* George F. Koster, John O. Dimmock, Robert G. Wheeler, Hermann Statz (1963). Properties of the thirty-two point groups. Published by the M.I.T. press, Cambridge, Massachusetts.
* Simon L. Altmann and Peter Herzig (1994). Point-Group Theory Tables. Oxford Science Publications.


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