Difference between revisions of "ApCoCoA-1:Modular group"

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(New page: === <div id="Modular_group">Modular group</div> === ==== Description ==== The Modular group has the following representation: PSL(2,Z) = <a,b | a...)
 
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   Relations:=CreateRelationsModular();
 
   Relations:=CreateRelationsModular();
 
   GB:=NC.GB(Relations);
 
   GB:=NC.GB(Relations);
 +
====Example in Symbolic Data Format====
 +
  <FREEALGEBRA createdAt="2014-01-24" createdBy="strohmeier">
 +
  <vars>a,b,d</vars>
 +
  <basis>
 +
  <ncpoly>b*d-1</ncpoly>
 +
  <ncpoly>d*b-1</ncpoly>
 +
  <ncpoly>a*a-1</ncpoly>
 +
  <ncpoly>(a*b)^3-1</ncpoly>
 +
  </basis>
 +
  <Comment>Modular_group</Comment>
 +
  </FREEALGEBRA>

Revision as of 17:58, 6 March 2014

Description

The Modular group has the following representation:

 PSL(2,Z) = <a,b | a^2 = (ab)^3 = 1 >

Reference

Platonic tilings of Riemann surfaces: The Modular Group, Gerard Westendorp

Computation

 /*Use the ApCoCoA package ncpoly.*/
 
 // a^{-1} is a and b is invers to d
 Use ZZ/(2)[a,b,d];
 NC.SetOrdering("LLEX");
 Define CreateRelationsModular()
   Relations:=[];
   // add the invers relations
   Append(Relations,[[b,d],[1]]);
   Append(Relations,[[d,b],[1]]);
   
   // add the relation a^2 = 1
   Append(Relations,[[a^2],[1]]);
   
   // add the relation (ab)^3 = 1
   Append(Relations,[[a,b,a,b,a,b],[1]]);
   
   Return Relations;
 EndDefine;
 
 Relations:=CreateRelationsModular();
 GB:=NC.GB(Relations);

Example in Symbolic Data Format

 <FREEALGEBRA createdAt="2014-01-24" createdBy="strohmeier">
 	<vars>a,b,d</vars>
 	<basis>
 	<ncpoly>b*d-1</ncpoly>
 	<ncpoly>d*b-1</ncpoly>
 	<ncpoly>a*a-1</ncpoly>
 	<ncpoly>(a*b)^3-1</ncpoly>
 	</basis>
 	<Comment>Modular_group</Comment>
 </FREEALGEBRA>