ApCoCoA-1:Dihedral groups

Dihedral Groups

Description

The dihedral group of degree n is the group of symmetries of a regular polynom. This non-abelian group consists of 2n elements, n rotations and n reflections. Let r be a single rotation and s be an arbitrary reflection. Then the group has the following representation

Dih(n) = <r,s | r^{n} = s^{2} = s^{-1}rs = r^{-1} = 1>

Reference

Reflection Groups and Invariant Theory, Richard Kane, Springer, 2001.

Computation

 /*Use the ApCoCoA package ncpoly.*/
 
 // Number of Dihedral group
 MEMORY.N:=5;
 
 Use ZZ/(2)[r,s];
 NC.SetOrdering("LLEX");
 
 Define CreateRelationsDihedral()
   Relations:=[];
   
   // add the relation r^{n} = 1    
   Append(Relations,[[r^MEMORY.N],[1]]);
   
   // add the relation s^2 = 1
   Append(Relations,[[s^2],[1]]);
   
   // add the relation s^{-1}rs = r^{-1}
   Append(Relations,[[s,r,s],[r^(MEMORY.N-1)]]);
   
   Return Relations;
 EndDefine;
 
 Relations:=CreateRelationsDihedral();
 Relations;
 
 Gb:=NC.GB(Relations);
 Gb;

Example in Symbolic Data Format

 <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
 	<vars>r,s</vars>
 	<basis>
 	<ncpoly>r^5-1</ncpoly>
 	<ncpoly>s*s-1</ncpoly>
 	<ncpoly>s*r*s-r^(5-1)</ncpoly>
 	</basis>
 	<Comment>Dihedral_group_5</Comment>
 </FREEALGEBRA>