Dimension1 Cyclic

The group $G=C_n$ has representation given by

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

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

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|
[1,0, 0,Zn^s]>;

For general $n$ and $s$ coprime to $n$, the Burnside symbol is $\begin{align*} [\mathbb{P}^1 \circlearrowleft G]=&(1,G \circlearrowright k(x),())\\ &+(C_n,1 \circlearrowright k,(s))+(C_n,1 \circlearrowright k,(n-s)) \end{align*}$

Kaiqi Yang(杨凯奇)