ApCoCoA:Modular group

From CoCoAWiki
Jump to: navigation, search

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>