Instructions

Given a representation $\Phi$ of a group $G \subset PGL_{n+1}$ acting on $\mathbb{P}^n$, follow the instructions to compute the Burnside symbols.

1. Copy all lines from Magma code.
2. Input 3 entries:
   • $FScale$, a cyclotomic field which contains eigenvalues of all elements in $\Phi(G)$.
   • $F$, extension of $FScale$, such that the representation of $\Phi(G)$ is defined over $F$.
   • $\Phi(G)$, a matrix group defined over $F$.
3. To compute Burnside symbols, use command

   
   BurnsideSymbols,PG2GHom,G2PGHom,PermG,H,LH,GrpQuo,AllGrp,ChainNode,ChainClass,
   
   ScalarGrp,AllSymbolNGLambda:=ComputeBurnsideSymbol(G,F,FScale);
   
   

   It computes the Burnside symbols, display the simplified version of Burnside symbols and computes the conjugated $\beta$ under the action of normalizer of stabilizer of the corresponding symbol.
4. To check the simplified version of BurnsideSymbols again, use command

   ReadBurnsideSymbols(BurnsideSymbols);
   

5. To check the conjugated $\beta$ again, use command

   EquivSymbols:=FindEquivSymbol(BurnsideSymbols,H);
   for i in [1..#EquivSymbols] do
   	EquivSymbols[i];
   end for;