Difference between revisions of "ApCoCoA-1:Dihedral groups"

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=== <div id="Dihedral groups">[[:ApCoCoA:Symbolic data#Dihedral_groups|Dihedral groups]]</div> ===
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=== <div id="Dihedral groups">[[:ApCoCoA:Symbolic data#Dihedral_groups|Dihedral Groups]]</div> ===
 
==== Description ====
 
==== 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
 
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
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   Use ZZ/(2)[r,s];
 
   Use ZZ/(2)[r,s];
 
   NC.SetOrdering("LLEX");
 
   NC.SetOrdering("LLEX");
   Define CreateRelationsDehidral()
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 +
   Define CreateRelationsDihedral()
 
     Relations:=[];
 
     Relations:=[];
 
      
 
      
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   EndDefine;
 
   EndDefine;
 
    
 
    
   Relations:=CreateRelationsDehidral();
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   Relations:=CreateRelationsDihedral();
 
   Relations;
 
   Relations;
 
    
 
    
   GB:=NC.GB(Relations);
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   Gb:=NC.GB(Relations);
   GB;
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   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>
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  <ncpoly>s*r*s-r^(5-1)</ncpoly>
 +
  </basis>
 +
  <Comment>Dihedral_group_5</Comment>
 +
  </FREEALGEBRA>

Latest revision as of 20:29, 22 April 2014

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>