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

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=== <div id="VonDyck groups">[[:ApCoCoA:Symbolic data#VonDyck_groups|Von Dyck groups]]</div> ===
+
=== <div id="VonDyck groups">[[:ApCoCoA:Symbolic data#VonDyck_groups|Von Dyck Groups]]</div> ===
 
==== Description ====
 
==== Description ====
 
+
The von Dyck groups are sometimes referred to as ordinary triangle groups and are subgroups of index 2 in Triangle(l, m, n) generated by words of even length in the generators a, b, c. A specific representation is given for x = ab, y = ca, yx = cb:
 
   D(l,m,n) = <x,y | x^{l} = y^{m} = (xy)^{n} = 1>
 
   D(l,m,n) = <x,y | x^{l} = y^{m} = (xy)^{n} = 1>
  
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   // Parameters of von Dyck group
 
   // Parameters of von Dyck group
 
 
 
   MEMORY.L:=3;
 
   MEMORY.L:=3;
 
   MEMORY.M:=5;
 
   MEMORY.M:=5;
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   Use ZZ/(2)[x,y];
 
   Use ZZ/(2)[x,y];
 
   NC.SetOrdering("LLEX");
 
   NC.SetOrdering("LLEX");
 +
 
 
   Define CreateRelationsVonDyck()
 
   Define CreateRelationsVonDyck()
 
     Relations:=[];
 
     Relations:=[];
 
      
 
      
 
     // add the relation x^l = 1
 
     // add the relation x^l = 1
    BufferA:=[];
 
 
     Append(Relations,[[x^MEMORY.L],[1]]);
 
     Append(Relations,[[x^MEMORY.L],[1]]);
 
      
 
      
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     Append(BufferXY,y);
 
     Append(BufferXY,y);
 
     EndFor;
 
     EndFor;
   
 
 
     Append(Relations,[BufferXY,[1]]);
 
     Append(Relations,[BufferXY,[1]]);
 
      
 
      
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   Relations:=CreateRelationsVonDyck();
 
   Relations:=CreateRelationsVonDyck();
 
   Relations;
 
   Relations;
   GB:=NC.GB(Relations);
+
    
   GB;
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  Gb:=NC.GB(Relations);
 +
   Gb;
 +
====Example in Symbolic Data Format====
 +
  <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
 +
  <vars>x,y</vars>
 +
  <basis>
 +
  <ncpoly>x^3-1</ncpoly>
 +
  <ncpoly>y^5-1</ncpoly>
 +
  <ncpoly>(x*y)^2-1</ncpoly>
 +
  </basis>
 +
  <Comment>Von_Dyck_group_l3m5n2</Comment>
 +
  </FREEALGEBRA>

Latest revision as of 20:45, 22 April 2014

Description

The von Dyck groups are sometimes referred to as ordinary triangle groups and are subgroups of index 2 in Triangle(l, m, n) generated by words of even length in the generators a, b, c. A specific representation is given for x = ab, y = ca, yx = cb:

 D(l,m,n) = <x,y | x^{l} = y^{m} = (xy)^{n} = 1>

(Reference: not found yet)

Computation

 /*Use the ApCoCoA package ncpoly.*/
 
 // Parameters of von Dyck group
 MEMORY.L:=3;
 MEMORY.M:=5;
 MEMORY.N:=2;
 
 Use ZZ/(2)[x,y];
 NC.SetOrdering("LLEX");
 
 Define CreateRelationsVonDyck()
   Relations:=[];
   
   // add the relation x^l = 1
   Append(Relations,[[x^MEMORY.L],[1]]);
   
   // add the relation y^m = 1
   Append(Relations,[[y^MEMORY.M],[1]]);
   
   // add the relation (xy)^n = 1
   BufferXY:=[];
   For Index1 := 1 To MEMORY.N Do
   	Append(BufferXY,x);
   	Append(BufferXY,y);
   EndFor;
   Append(Relations,[BufferXY,[1]]);
   
   Return Relations;
 EndDefine;
 
 Relations:=CreateRelationsVonDyck();
 Relations;
 
 Gb:=NC.GB(Relations);
 Gb;

Example in Symbolic Data Format

 <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
 	<vars>x,y</vars>
 	<basis>
 	<ncpoly>x^3-1</ncpoly>
 	<ncpoly>y^5-1</ncpoly>
 	<ncpoly>(x*y)^2-1</ncpoly>
 	</basis>
 	<Comment>Von_Dyck_group_l3m5n2</Comment>
 </FREEALGEBRA>