Dimension1 Dihedral

When $n$ is odd, group $G=D_n$ has representation given by

\[\begin{align*} G=\langle \begin{pmatrix} \zeta_n^s & 0 \\ 0 & \zeta_n^{-s} \end{pmatrix}, \begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix} \rangle, \end{align*}\]

where $\zeta_n$ is a $n$-th root of unity, $s$ is coprime to $n$.

When $n$ is even, group $G=D_n$ has projective representation given by

\[\begin{align*} G=\langle \begin{pmatrix} \zeta_{2n}^s & 0 \\ 0 & \zeta_{2n}^{-s} \end{pmatrix}, \begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix} \rangle, \end{align*}\]

where $\zeta_{2n}$ is a $2n$-th root of unity, $s$ is coprime to $2n$.

Here we take $n=3,s=1$ as an example, the Magma code is as follows:


n:=3;
s:=1;

FScale:=CyclotomicField(n:Sparse:=true);

F:=FScale;

Zn:=RootOfUnity(n);

G:=MatrixGroup<2,F|
[Zn^s,0, 0,Zn^(-s)],
[0,1, 1,0]>;

If $n$ is odd and $s$ coprime to $n$, the Burnside symbol is

\[\begin{align*} [\mathbb{P}^1 \circlearrowleft G]=&(1,G \circlearrowright k(x),())\\ &+2(C_2,1 \circlearrowright k,(1))\\ &+(C_n,1 \circlearrowright k,(s)) \end{align*}\]

where

\[\begin{align*} C_2=\langle \begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix}\rangle, C_n=\langle \begin{pmatrix} \zeta_n^s & 0\\ 0 & \zeta_n^{-s} \end{pmatrix} \rangle \end{align*}\]

If $n$ is even and $s$ coprime to $2n$, the Burnside symbol is

\[\begin{align*} [\mathbb{P}^1 \circlearrowleft G]=&(1,G \circlearrowright k(x),())\\ &+(1.C_2,1 \circlearrowright k,(1))+(2.C_2,1 \circlearrowright k,(1))\\ &+(C_n,1 \circlearrowright k,(s)) \end{align*}\]

where

\[\begin{align*} 1.C_2=\langle \begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix}\rangle, 2.C_2=\langle \begin{pmatrix} 0 & i\\ -i & 0 \end{pmatrix}\rangle, C_n=\langle \begin{pmatrix} \zeta_{2n}^s & 0\\ 0 & \zeta_{2n}^{-s} \end{pmatrix} \rangle \end{align*}\]

Kaiqi Yang(杨凯奇)