I am trying to write down a representation of $D_n = \langle \sigma, \tau \mid \sigma^n = \tau^2 =e, \tau \sigma = \sigma^{-1} \tau \rangle$ over $\mathbb{R}^2 \cong \mathbb{C}$ (as an $\mathbb{R}$-vector space).

What I Know:

My representation has to be a group homomorphism $\rho: D_n \rightarrow \text{GL}(2, \mathbb{R})$. I know that the homomorphism will be fully specified by specifying where I send the generators of $D_n$. Intuitively I know that I should be sending $\sigma$ to the $2 \times 2 $ rotation matrix for an angle of $2 \pi /n$ and $\tau$ to a horizontal reflection.

This makes me think I should be choosing \begin{array}{c c c} \sigma^k \rightarrow \begin{pmatrix} \cos{(2\pi k /n)} & -\sin{(2\pi k /n)} \\ \sin{(2\pi k /n)} & \cos{(2\pi k /n)} \end{pmatrix} & \text{and }& \tau \rightarrow \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \\ \end{array}

My Problem:

I am having issues checking that this map respects the group operation. I know I want to check that: $$ \rho((\sigma^a \tau^b)(\sigma^c\tau^d)) = \rho(\sigma^a \tau^b) \rho(\sigma^c\tau^d) $$ since an arbitrary member of $D_n$ can be written as $\sigma^a \tau^b$. I think it would suffice to show that $\rho(\sigma \tau)= \rho(\sigma) \rho(\tau)$. I can easily write down the matrices on the RHS as I have just defined them above. However, I don't know how to check to see if this product matrix equals the matrix for $\rho(\sigma \tau)$, since I don't know what that is! How should I go about verifying my homomorphism is in fact a homomorphism?

  • $\begingroup$ I think this answer should help. The gist is that, to verify your map is a homomorphism, you simply need to make sure that the relations defining $D_n$ hold for $\rho(D_n)$; e.g., that $\rho(\sigma)^n = 1$, that $\rho(\sigma\tau\sigma\tau) = 1$, etc. I know this sort of thing was (is) a gap in my group theory knowledge... $\endgroup$ – pjs36 Jan 25 '18 at 4:19

You can avoid a lot of work by just using the presentation of $D_n$. The presentation of $D_n$ means that given a group $G$ and elements $s,t\in G$ such that $s^n=t^2=e$ and $ts=s^{-1}t$, there is a unique homomorphism $\rho:D_n\to G$ such that $\rho(\sigma)=s$ and $\rho(\tau)=t$. In this case, $G=GL(2,\mathbb{R})$, so you just have to check that the matrices $s$ and $t$ which you want to be $\rho(\sigma)$ and $\rho(\tau)$ satisfy the relations.

Alternatively, if you really want to compute everything explicitly, here's what you can do. The trick is to choose some specific way of writing each element of $D_n$ and use that to define $\rho$. For instance, every element of $D_n$ can be written uniquely in the form $\sigma^a\tau^b$ where $a\in\{0,1,\dots,n-1\}$ and $b\in \{0,1\}$. So use this as your definition: define $\rho(\sigma^a\tau^b)=\rho(\sigma)^a\rho(\tau)^b$ when $a\in\{0,1,\dots,n-1\}$ and $b\in \{0,1\}$, where $\rho(\sigma)$ and $\rho(\tau)$ are the two matrices you chose. This uniquely defines $\rho$ on every element of $D_n$. Now you just have to check that this is a homomorphism, in other words that $$\rho((\sigma^a \tau^b)(\sigma^c\tau^d)) = \rho(\sigma^a \tau^b) \rho(\sigma^c\tau^d)$$ whenever $a,c\in\{0,1,\dots,n-1\}$ and $b,d\in \{0,1\}$. To check this, you first need to write $(\sigma^a \tau^b)(\sigma^c\tau^d)$ in the form $\sigma^e\tau^f$ so that you can evaluate $\rho$ on it using the definition above. Once you have that, you just have to compute the matrices and check they are equal.

  • $\begingroup$ Perhaps this is trivial, but why should $\rho$ in your first paragraph be unique? $\endgroup$ – gabe Jan 25 '18 at 6:02
  • $\begingroup$ That is part of the definition of a group presentation. If you like, it is because $\sigma$ and $\tau$ generate $D_n$. $\endgroup$ – Eric Wofsey Jan 25 '18 at 6:13
  • $\begingroup$ Thanks, this is very helpful. $\endgroup$ – gabe Jan 25 '18 at 6:43

All you have to do is set $$\rho(\sigma)=\pmatrix{\cos (2\pi/n)&-\sin(2\pi/n)\\\sin(2\pi/n)&\cos(2\pi/n)}$$ and $$\rho(\tau)=\pmatrix{1&0\\0&-1}$$ and then verify these preserve the relations, that is $\rho(\sigma)^n=I$ (standard property of rotation matrices), $\rho(\tau)^2=I$ (obvious) and $\rho(\tau)\rho(\sigma)=\rho(\sigma)^{-1}\rho(\tau)$ (a couple of lines of calculation).

Anyway these are just the matrices representing the usual action of $D_n$ as the symmetry group of the regular $n$-gon with centre at the origin.

  • $\begingroup$ Thanks, I didn't realize showing that if the imagine of sigma (tau) obeys the same relations as sigma (tau), I have a homomorphism. $\endgroup$ – gabe Jan 25 '18 at 6:44

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.