2
$\begingroup$

Currently, I am a bit confused by two representations of the dihedral group. In my search for an answer, I came across this unanswered question, which explains these two representation far more beautiful than I can. Here, it is suggested that there actually is no isomorphism between these two representations. But how can that be the case when they describe the same symmetry?

Or does there exists an isomorphism between these two representations? And if so what would this isomorphism be? (My attempts fail due to the fact that after flipping the rotations will be in different directions in the two representations.)

$\endgroup$
3
  • 1
    $\begingroup$ Yes, they are isomorphic; they define the same thing: that's what an isomorphism is for! $\endgroup$
    – Shaun
    Aug 26 at 21:27
  • $\begingroup$ But what exactly would this isomorphism be? I have been trying to work it out for the case of $D_8$, but I have been getting stuck when it comes to associating actual elements. Clearly, due to order relations, reflections and rotations need to be associated with each other respectively, but things break when I try to apply a reflection after a rotation, because the squares turn in opposite directions. $\endgroup$
    – Yadeses
    Aug 26 at 21:34
  • $\begingroup$ It sounds like you're having a left-right issue. You can either compose functions from right to left or from left to right. Both are allowed. Try mapping the 90 degree rotation in the one viewpoint to the 90 degree rotation in the other direction, keeping reflections sent to the same reflections. See if that fixes the problem. $\endgroup$ Aug 26 at 23:30
2
$\begingroup$

I think the question that you reference is somewhat unclear. I think the confusion is between (1) describing a symmetry $\theta$ of a regular polygon by describing a sequence of simple symmetries (rotations and reflections) that result in $\theta$, as opposed (2) to describing it by numbering the vertices in its initial position and then giving the numbering of the vertices in each position after applying the symmetry $\theta$. (I have a recollection of a blog by Terence Tao that compared these two approaches, but unfortunately I can't locate it just now.)

In approach (1), it is clear how to get the description of two symmetries given the descriptions of the two symmetries - you just concatenate the sequences of simple symmetries. In approach (2), it is not. To see why approach (2) is actually equivalent to approach (1) and to see how to get the approach (2) descriptions of the composition of two symmetries described using approach (2), you can first condense the approach (2) descriptions by imagining that you were initially standing at vertex 1 facing clockwise. An approach (2) description is then determined by a statement of the form "I am now at vertex $i$ facing clockwise" or "I am now at vertex $i$ facing anticlockwise". In this simplified version of approach (2), it should be clear that you get the descriptions of the composition of the symmetries by taking the difference of the vertex positions and (in a suitable sense) of the directions. Each of the approach (1) descriptions has an equivalent approach (2) description and vice versa, so the two methods of description are equivalent.

$\endgroup$
4
  • $\begingroup$ Okay, yes that makes a lot of sense. I suppose that is connected to how after a reflection, rotations like $(12\dots n)$ go in the other direction. I think the final point I still have to figure out/convince myself of is why viewpoint (1) and (2) endup back in the same place after one does a second reflection. I think this is connected to the words "compose the descriptions". But I am not sure I am fully grasping those words. $\endgroup$
    – Yadeses
    Aug 26 at 23:05
  • $\begingroup$ At the expense of making it even more verbose, I have tried to clarify my somewhat loose phraseology. I hope that helps. $\endgroup$
    – Rob Arthan
    Aug 26 at 23:10
  • $\begingroup$ For future reference: what helped me fully grasp the difference between these two approaches in the end was by working out $srs = r^{-1}$ in both approaches with drawings. $\endgroup$
    – Yadeses
    Aug 26 at 23:30
  • $\begingroup$ Yes, a picture is worth a thousand words - particularly a picture that you have drawn for yourself to understand a problem in mathematics $\ddot{\smile}$. $\endgroup$
    – Rob Arthan
    Aug 26 at 23:32

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.