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3 Dimensional Representation of D3

  1. Dec 2, 2014 #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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  3. Dec 2, 2014 #2

    pasmith

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    [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.
     
  4. Dec 2, 2014 #3

    ChrisVer

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    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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