3 Dimensional Representation of D3

In summary, the conversation discusses the process of obtaining a three dimensional representation of the Dihedral group of order 6, D_3, which has elements corresponding to rotations and reflections on the xy plane. It is noted that D_3 is isomorphic to S_3 and can be represented in \mathbb{R}^3 by permuting the standard basis vectors, but there is some confusion about how the rotations and reflections would fit into this representation.
  • #1
ChrisVer
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I was wondering how can I obtain the three dimensional representation of the Dihedral group of order 6, [itex]D_3[/itex].

If this group has the elements: [itex]D_3 = \left \{ e,c,c^2,b,bc,bc^2 \right \}[/itex]

Where [itex]c[/itex] corresponds to rotation by [itex]120^o[/itex] on the xy plane (so about z-axis) and [itex]b[/itex] to reflections of the [itex]x[/itex] axis, I don't see how [itex]z[/itex] would change at all... so when I try to obtain it I'm getting the known 2-dimensional representation of [itex]D_3[/itex] together with an extra [itex]1[/itex] on the diagonal corresponding to the transformations [itex]z \rightarrow z'=z[/itex]...
Any help?
 
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  • #2
[itex]D_3[/itex] is isomorphic to [itex]S_3[/itex], which acts naturally (but reducibly) on [itex]\mathbb{R}^3[/itex] by permuting the standard basis vectors.
 
  • #3
So you are suggesting it will have the 3dim repr of S3?
I also thought that... but then the rotations and reflections as defined don't seem to help in this correspondence...:s
 

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