ApCoCoA-1:SpecialLinear group
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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>