Difference between revisions of "ApCoCoA-1:VonDyck groups"
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(New page: === <div id="VonDyck groups">Von Dyck groups</div> === ==== Description ==== D(l,m,n) = <x,y | x^{l} = y^{m} = (xy)^{n} = 1> (Reference: not f...) |
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| − | === <div id="VonDyck groups">[[:ApCoCoA:Symbolic data#VonDyck_groups|Von Dyck | + | === <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> | ||
| Line 10: | Line 10: | ||
/*Use the ApCoCoA package ncpoly.*/ | /*Use the ApCoCoA package ncpoly.*/ | ||
| − | // | + | // Parameters of von Dyck group |
| − | MEMORY.N:= | + | MEMORY.L:=3; |
| + | MEMORY.M:=5; | ||
| + | MEMORY.N:=2; | ||
| + | Use ZZ/(2)[x,y]; | ||
| + | NC.SetOrdering("LLEX"); | ||
| − | + | Define CreateRelationsVonDyck() | |
| − | |||
| − | Define | ||
Relations:=[]; | Relations:=[]; | ||
| − | |||
| − | |||
| − | |||
| − | // add the relation | + | // add the relation x^l = 1 |
| − | Append(Relations, [[ | + | 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; | Return Relations; | ||
EndDefine; | EndDefine; | ||
| − | Relations:= | + | Relations:=CreateRelationsVonDyck(); |
Relations; | 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> | ||
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>
