We have studied the lepton and quark mixing patterns which can be derived from the
dihedral group

Article funded by SCOAP3

1$. A regular polygon with $n$ sides has $2n$ different symmetries: $n$ rotational symmetries and $n$ reflection symmetries, therefore the group order of $D_{n}$ is $2n$. All $D_{n}$ are non-ablelian permutation groups for $n>2$, $D_{1}$ is isomorphic to $Z_{2}$ and $D_{2}$ is isomorphic to $Z_{2}\times Z_{2}$. The group $D_{n}$ is the semidirect product $Z_{n}\rtimes Z_{2}$ of the cyclic groups $Z_{n}$ and $Z_{2}$. The dihedral group can be conveniently defined by two generators $R$ and $S$ which obey the relations~\cite{Blum-ml-2007jz,lomont1959applications,Georgi-ml-1999wka}, \begin{equation} \label{eq:Dn_multi_rules} R^{n}=S^{2}=(RS)^2=1\,, \end{equation} where $R$ refers to rotation and $S$ is the reflection. As a consequence, all the group elements of $D_{n}$ can be expressed as \begin{equation} g=S^{\alpha}R^{\beta}\, \end{equation} \looseness=-1 where $\alpha=0, 1$ and $\beta=0, 1, \dots, n-1$. Then it is straightforward to determine the conjugacy classes of the dihedral group. Depending on whether the group index $n$ is even or odd, the $2n$ group elements of $D_n$ can be classified into three or five types of conjugacy classes. \begin{itemize} \item{$n$ is odd} \begin{equation} \begin{aligned} 1C_{1}&=\{1\}\,,\\ 2C_{m}^{(\rho)}&=\{R^{\rho},R^{-\rho}\}\,,~~\text{with}~~\rho=1,\dots,\frac{n-1}{2}\,,\\ nC_{2}&=\{S,SR,SR^{2},\dots,SR^{n-1}\}\,, \end{aligned} \end{equation} where $m$ is minimal integer such that the identity $m\rho=0 \,(\mathrm{mod}~n)$ is satisfied, and $kC_l$ denotes a conjugacy class of $k$ elements whose order are $l$. \item{$n$ is even} \begin{equation} \begin{aligned} 1C_{1}&=\{1\}\,,\\ 1C_{2}&=\{R^{n/2}\}\,,\\ 2C_{m}^{(\rho)}&=\{R^{\rho},R^{-\rho}\}\,,~~\text{with}~~\rho=1,\dots,\frac{n-2}{2}\,,\\ \frac{n}{2}C_{2}&=\{S,SR^{2},SR^{4},\dots,SR^{n-4},SR^{n-2}\}\,,\\ \frac{n}{2}C_{2}&=\{SR,SR^{3},\dots,SR^{n-3},SR^{n-1}\}\,, \end{aligned} \end{equation} \end{itemize} The group structure of $D_n$ is simple, and the subgroups of $D_{n}$ group turn out to be either dihedral or cyclic group. The explicit expressions of all the subgroups are \begin{align} \nonumber Z_{j}&= \langle R^{\frac{n}{j}}\rangle \,& \text{with}~~j|n&\,,\\ \nonumber Z_{2}^{(m)}&=\langle SR^{m}\rangle \,& \text{with}~~m&=0,1,\dots,n-1\,,\\ D_{j}^{(m)}&=\langle R^{\frac{n}{j}},SR^{m}\rangle \,&\text{with}~~j|n,~ m&=0,1,\dots,\frac{n}{j}-1\,. \end{align} Hence the total number of cyclic subgroups generated by certain power of $R$ is equal to the number of positive divisors of $n$, and the total number of dihedral subgroups is the sum of positive divisors of $n$. The group $D_{n}$ only has real one-dimensional and two-dimensional irreducible representations. The number of irreducible representations is dependent on the parity of the group index $n$. \begin{itemize} \item{$n$ is odd} If the index $n$ is an odd integer, the group $D_{n}$ has two singlet representations $\mathbf{1}_{i}$ and $\frac{n-1}{2}$ doublet representations $\mathbf{2}_{j}$, where the indices $i$ and $j$ are $i=1, 2$ and $j=1,\dots,\frac{n-1}{2}$. We observe that the sum of the squares of the dimensions of the irreducible representations is \begin{equation} 1^2+1^2+2^{2}\times \frac{n-1}{2}=2n\,, \end{equation} which is exactly the number of elements in $D_n$ group. In the one-dimensional representations, we have \begin{equation} \mathbf{1}_{1}:~R=S=1\,, \quad \mathbf{1}_{2}:~R=1,~S=-1\,. \end{equation} For the two-dimensional representations, the generators $R$ and $S$ are represented by \begin{equation} \mathbf{2}_{j}:~R=\left(\begin{array}{cc} e^{2\pi i\frac{j}{n}} & 0 \\ 0 & e^{-2\pi i\frac{j}{n}} \end{array} \right)\,, \quad S=\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)\,, \end{equation} with $j=1,\dots,\frac{n-1}{2}$. \item{$n$ is even} For the case that the index $n$ is an even integer, the group $D_{n}$ has four singlet representations $\mathbf{1}_{i}$ with $i=1,2,3,4$ and $\frac{n}{2}-1$ doublet representations $\mathbf{2}_{j}$ with $j=1,\dots,\frac{n}{2}-1$. It can be checked that the squared dimensions of the inequivalent irreducible representations add up to the group order as well, \begin{equation} 1^2+1^2+1^2+1^2+2^{2} \times \left(\frac{n}{2}-1\right)=2n\,. \end{equation} The generators $R$ and $S$ for the one-dimensional representations are given by \begin{equation} \begin{aligned} \mathbf{1}_{1}:~R&=S=1\,, &\mathbf{1}_{2}:~R&=1,~S=-1\,,\\ \mathbf{1}_{3}:~R&=-1,~S=1\,, & \mathbf{1}_{4}:~R&=S=-1\,. \end{aligned} \end{equation} The explicit forms of these generators in the irreducible two-dimensional representations are \begin{equation} \mathbf{2}_{j}:~R=\left(\begin{array}{cc} e^{2\pi i\frac{j}{n}} & 0 \\ 0 & e^{-2\pi i\frac{j}{n}} \end{array} \right)\,, \quad S=\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)\,, \end{equation} with $j=1,\dots,\frac{n}{2}-1$. Notice that the doublet representation $\mathbf{2}_{j}$ and the complex conjugate $\bar{\mathbf{2}}_{j}$ are unitarily equivalent, and they are related through change of basis, i.e., $R^{*}=URU^{-1}$ and $S^{*}=USU^{-1}$ where $U=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$. Hence all the two-dimensional representations of $D_n$ are real representations, although the representation matrix of $R$ is complex in the chosen basis. Moreover, if $a=\left(a_1, a_2\right)^{T}$ is a doublet transforming as $\mathbf{2}_j$, the complex conjugate $\bar{a}=\left(a^{*}_1, a^{*}_2\right)^{T}$ doesn't transform as $\mathbf{2}_j$, but rather $\left(a^{*}_2, a^{*}_1\right)^{T}$ transform as $\mathbf{2}_j$ under $D_n$. \end{itemize} In order to consistently combine a flavor symmetry $G_f$ with the CP symmetry, the subsequent action of the CP transformation, an element of the flavor group and the CP transformation should be equivalent to the action of another element of the flavor group. In other word, the so-called consistency condition has to be fulfilled~\cite{Grimus-ml-1995zi,Feruglio-ml-2012cw,Holthausen-ml-2012dk,Chen-ml-2014tpa} \begin{equation} \label{eq:consis_cond}X_{\mathbf{r}}\rho^{*}_{\mathbf{r}}(g)X^{\dagger}_{\mathbf{r}}=\rho_{\mathbf{r}}(g^{\prime}),\quad g,g^{\prime}\in G_{f}\,, \end{equation} where $\rho_{\mathbf{r}}(g)$ is the representation matrix of the element $g$ in the representation $\mathbf{r}$, and $X_{\mathbf{r}}$ is the CP transformation. Here $g$ and $g'$ are generally different group elements, consequently the CP transformation $X_{\bf r}$ is related to an automorphism which maps $g$ into $g^{\prime}$. In addition, ref.~\cite{Chen-ml-2014tpa} showed that physical CP transformations should be a class-inverting automorphism of $G_f$, i.e.\ $g^{-1}$ and $g'$ which is the image of $g$ under the automorphism should be in the same conjugacy class. We find that the $D_n$ groups really have a class-inverting outer automorphism $\mathfrak{u}$, and its action on the generators is \begin{equation} \label{eq:class_inverting_aut_Dn} R\stackrel{\mathfrak{u}}{\longmapsto}R^{-1}\,,\quad S\stackrel{\mathfrak{u}}{\longmapsto}S\,. \end{equation} The CP transformation corresponding to $\mathfrak{u}$ is denoted by $X^0_{\mathbf{r}}$, its concrete form is determined by the following consistency conditions, \begin{eqnarray} \nonumber X^0_{\mathbf{r}}\rho^{*}_{\mathbf{r}}(R)X^{0\dagger}_{\mathbf{r}}&=&\rho_{\mathbf{r}}\left(\mathfrak{u}\left(R\right)\right)=\rho_{\mathbf{r}}\left(R^{-1}\right)\,,\\ \label{eq:cons_eq} X^0_{\mathbf{r}}\rho^{*}_{\mathbf{r}}(S)X^{0\dagger}_{\mathbf{r}}&=&\rho_{\mathbf{r}}\left(\mathfrak{u}\left(S\right)\right)=\rho_{\mathbf{r}}\left(S\right)\,. \end{eqnarray} In our working basis shown above, $X^0_{\mathbf{r}}$ is fixed to be a unit matrix up to an overall irrelevant phase, \begin{equation} \label{eq:gcp_trans}X^0_{\mathbf{r}}=\mathbb{1}\,. \end{equation} Furthermore, including the inner automorphisms, the full set of CP transformations compatible with $D_{n}$ flavor symmetry are \begin{equation} \label{eq:GCP_all}X_{\mathbf{r}}=\rho_{\mathbf{r}}(g)X^0_{\mathbf{r}}=\rho_{\mathbf{r}}(g),\quad g\in D_{n}\,. \end{equation} Hence the CP transformations compatible with $D_n$ are of the same form as the flavor symmetry transformations in the chosen basis. ]]>