I have a problem doing Exercise 18.8 of groups and symmetry by M. A. Armstrong.
18.8. Look at our description of the Möbius band as a subset of $\mathbb{C} \times \mathbb{C}$ and find matrices in $U_2$ which represent the symmetries $r$ and $s$.
The symmetries $r,s$ are defined in the text:
A $5 \times 1$ rectangular strip of paper is marked off on both sides into five unit squares. The ends of the strip are then joined with a half twist to produce a Möbius band $M$.
Make a model of $M$ and run it through your fingers so that the squares move along one position. Call this symmetry $r$. After ten moves you get back to where you started, so $r^{10}$ is the identity. (Note that $r$ is not induced by a rotation of $\mathbb{R}^3$; it is a movement of the Möbius band in itself. This type of symmetry shows up well if the band is used as a belt drive connecting two pulleys.)
There is another natural symmetry $s$; just turn $M$ over as in Figure 18.3. Together $r$ and $s$ generate a group which is isomorphic to the dihedral group $D_{10}$ and which acts on the set of painted bands in the obvious way. The conjugacy classes of $D_{10}$ are \begin{gathered} \{e\}, \quad\left\{r, r^9\right\}, \quad\left\{r^2, r^8\right\}, \\ \left\{r^3, r^7\right\}, \quad\left\{r^4, r^6\right\}, \quad\left\{r^5\right\} \\ \left\{s, r^2 s, r^4 s, r^6 s, r^8 s\right\}, \end{gathered} and $$ \left\{r s, r^3 s, r^5 s, r^7 s, r^9 s\right\} $$ The reader may well wish to have explicit formulae for the symmetries $r$ and $s$. The neatest approach is as follows. Think of $M$ as the subset $$ \left\{\left(e^{2 i \theta}, \lambda e^{i \theta}\right) \middle|-\pi<\theta \leqslant \pi, 0 \leqslant \lambda \leqslant 1\right\} $$ of $\mathbb{C} \times \mathbb{C}$. Then $r$ sends $\left(e^{2 i \theta}, \lambda e^{i \theta}\right)$ to $\left(e^{2 i(\theta+\pi / 5)}, \lambda e^{i(\theta+\pi / 5)}\right)$, and $s$ sends $\left(e^{2 i \theta}, \lambda e^{i \theta}\right)$ to $\left(e^{-2 i \theta}, \lambda e^{-i \theta}\right)$.
From the above I know that Möbius band is a subset of $\Bbb C\times\Bbb C$ with symmetry group $D_{10}$ (the symmetry group of 10-gon) generated by $r,s$ with the relations $r^{10}=s^2=rsrs=e$ and Exercise 18.8 asks us to embed $D_{10}$ in $U_2$. ($U_2$ consists of complex unitary matrixes of order $2$. An invertible matrix $U$ is unitary iff its conjugate transpose $U^*$ equals $U^{-1}$).
My attempt:
Since $r$ sends $\left(e^{2 i \theta}, \lambda e^{i \theta}\right)$ to $\left(e^{2 i(\theta+\frac\pi5)}, \lambda e^{i(\theta+\frac\pi5)}\right)$, it is represented by\begin{bmatrix}e^{2i\pi\over5}\\&e^{i\pi\over5}\end{bmatrix} I can't find a unitary matrix representing $s$: $$\left(e^{2 i \theta}, \lambda e^{i \theta}\right)\mapsto\left(e^{-2 i \theta}, \lambda e^{-i \theta}\right)$$ In other words $s$ conjugates each coordinate, which is $\Bbb R$-linear, represented by an element of $O_4$:\begin{bmatrix}1\\&-1\\&&1\\&&&-1\end{bmatrix} I think $s$ is not $\Bbb C$-linear (as $k\overline1\ne\overline{k1}$ for $k\notin\Bbb R$), so I can't find an element of $U_2$ representing $s$.