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Point Group Tables of D3h(-6m2)

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Character Table of the group D3h(-6m2)*
D3h(-6m2)#1m3-62120m100functions
Mult.-112233·
A'1Γ1111111x2+y2,z2
A'2Γ21111-1-1Jz
A''1Γ31-11-11-1·
A''2Γ41-11-1-11z
E'Γ622-1-100(x,y),(x2-y2,xy)
E''Γ52-2-1100(xz,yz),(Jx,Jy)




[ Subduction tables ]

Multiplication Table of irreducible representations of the group D3h(-6m2)
D3h(-6m2)A'1A'2A''1A''2E'E''
A'1A'1A'2A''1A''2E'E''
A'2·A'1A''2A''1E'E''
A''1··A'1A'2E''E'
A''2···A'1E''E'
E'····A'1+A'2+E'A''1+A''2+E''
E''·····A'1+A'2+E'

[ Note: the table is symmetric ]


Symmetrized Products of Irreps
D3h(-6m2)A'1A'2A''1A''2E'E''
[A'1 x A'1]1·····
[A'2 x A'2]1·····
[A''1 x A''1]1·····
[A''2 x A''2]1·····
[E' x E']1···1·
[E'' x E'']1···1·


Antisymmetrized Products of Irreps
D3h(-6m2)A'1A'2A''1A''2E'E''
{A'1 x A'1}······
{A'2 x A'2}······
{A''1 x A''1}······
{A''2 x A''2}······
{E' x E'}·1····
{E'' x E''}·1····


Irreps Decompositions
D3h(-6m2)A'1A'2A''1A''2E'E''
V···11·
[V2]2···11
[V3]11·221
[V4]3·1132
A·1···1
[A2]2···11
[A3]·21112
[A4]3·1132
[V2]xV111342
[[V2]2]5·1143
{V2}·1···1
{A2}·1···1
{[V2]2}121123

V ≡ the vector representation
A ≡ the axial representation


IR Selection Rules
IRA'1A'2A''1A''2E'E''
A'1···xx·
A'2··x·x·
A''1·x···x
A''2x····x
E'xx··xx
E''··xxxx

[ Note: x means allowed ]


Raman Selection Rules
RamanA'1A'2A''1A''2E'E''
A'1x···xx
A'2·x··xx
A''1··x·xx
A''2···xxx
E'xxxxxx
E''xxxxxx

[ Note: x means allowed ]


Irreps Dimensions Irreps of the point group
Subduction of the rotation group D(L) to irreps of the group D3h(-6m2)
L2L+1A'1A'2A''1A''2E'E''
011·····
13···11·
251···11
3711·111
491·1121
51111·122
613211122
715111232
817211133
919221233
1021212243


* 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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