ApCoCoA-1:SpecialLinear group

Special Linear Group

Description

The special linear group over Z is the group of all n x n matrices with coefficients in Z which has a determinant equals 0. A simple representation is given by:

 SL_2(Z) = <a,b | aba = bab, (aba)^4 = 1 >

Reference

Andrew Baker: An introduction to matrix groups and their applications

Computation

 /*Use the ApCoCoA package ncpoly.*/
 
 // a is invers to c, b is invers to d
 Use ZZ/(2)[a,b,c,d];
 NC.SetOrdering("LLEX");
 Define CreateRelationsSpecialLinear()
   Relations:=[];
 
   // add the invers relations
   Append(Relations,[[a,c],[1]]);
   Append(Relations,[[c,a],[1]]);
   Append(Relations,[[b,d],[1]]);
   Append(Relations,[[d,b],[1]]);
  
   // add the relation aba = bab 
   Append(Relations,[[a,b,a],[b,a,b]]);
   
   // add the relation (aba)^4 = 1
   Append(Relations,[[a,b,a,a,b,a,a,b,a,a,b,a],[1]]);
  
   Return Relations;
 EndDefine;
 
 Relations:=CreateRelationsSpecialLinear();
 Gb:=NC.GB(Relations,31,1,100,1000);

Example in Symbolic Data Format

 <FREEALGEBRA createdAt="2014-01-20" createdBy="strohmeier">
 	<vars>a,b,c,d</vars>
 	<uptoDeg>9</uptoDeg>
 	<basis>
 	<ncpoly>a*c-1</ncpoly>
 	<ncpoly>c*a-1</ncpoly>
 	<ncpoly>b*d-1</ncpoly>
 	<ncpoly>d*b-1</ncpoly>
 	<ncpoly>a*b*a-b*a*b</ncpoly>
 	<ncpoly>(a*b*a)^4-1</ncpoly>
 	</basis>
 	<Comment>Special_Linear_group</Comment>
 </FREEALGEBRA>