ApCoCoA:Hecke groups

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Description

The Hecke groups has the following representation:

 H(lambda_q) = <x,y | x^2=(xy)^q=1, for q >= 3>

Reference

GROWTH IN HECKE GROUPS, MARTIN KREUZER AND GERHARD ROSENBERGER

Computation

 /*Use the ApCoCoA package ncpoly.*/
 
 // Define the variable q of the Hecke group
 MEMORY.Q := 4;
 
 // y is invers to z, x is invers to itself (that follows directly from the first relation)
 Use ZZ/(2)[x,y,z];
 NC.SetOrdering("LLEX");
 
 Define CreateRelationsHecke()
   Relations:=[];
   // add the invers relation of y and z
   Append(Relations,[[y,z],[1]]);
   Append(Relations,[[z,y],[1]]);
   
   // add the relation x^2 = 1
   Append(Relations,[[x,x],[1]]);
   
   // add the relation (xy)^q = 1
   RelationBuffer1:=[];
   For Index0 := 1 To MEMORY.Q Do
     Append(RelationBuffer1,x);
     Append(RelationBuffer1,y);
   EndFor;
   Append(Relations,[RelationBuffer1,[1]]);
 	
   Return Relations;
 EndDefine;
 
 Relations:=CreateRelationsHecke();
 Gb:=NC.GB(Relations);

Examples in Symbolic Data Format

Hecke group 4
 <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
 	<vars>x,y,z</vars>
 	<basis>
 	<ncpoly>y*z-1</ncpoly>
 	<ncpoly>z*y-1</ncpoly>
 	<ncpoly>x*x-1</ncpoly>
 	<ncpoly>((x*y)^4)-1</ncpoly>
 	</basis>
 	<Comment>Hecke_group_4</Comment>
 </FREEALGEBRA>
Hecke group 5
 <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
 	<vars>x,y,z</vars>
 	<basis>
 	<ncpoly>y*z-1</ncpoly>
 	<ncpoly>z*y-1</ncpoly>
 	<ncpoly>x*x-1</ncpoly>
 	<ncpoly>((x*y)^5)-1</ncpoly>
 	</basis>
 	<Comment>Hecke_group_5</Comment>
 </FREEALGEBRA>
Hecke group 6
 <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
 	<vars>x,y,z</vars>
 	<basis>
 	<ncpoly>y*z-1</ncpoly>
 	<ncpoly>z*y-1</ncpoly>
 	<ncpoly>x*x-1</ncpoly>
 	<ncpoly>((x*y)^6)-1</ncpoly>
 	</basis>
 	<Comment>Hecke_group_6</Comment>
 </FREEALGEBRA>
Hecke group 7
 <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
 	<vars>x,y,z</vars>
 	<basis>
 	<ncpoly>y*z-1</ncpoly>
 	<ncpoly>z*y-1</ncpoly>
 	<ncpoly>x*x-1</ncpoly>
 	<ncpoly>((x*y)^7)-1</ncpoly>
 	</basis>
 	<Comment>Hecke_group_7</Comment>
 </FREEALGEBRA>
Hecke group 8
 <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
 	<vars>x,y,z</vars>
 	<basis>
 	<ncpoly>y*z-1</ncpoly>
 	<ncpoly>z*y-1</ncpoly>
 	<ncpoly>x*x-1</ncpoly>
 	<ncpoly>((x*y)^8)-1</ncpoly>
 	</basis>
 	<Comment>Hecke_group_8</Comment>
 </FREEALGEBRA>
Hecke group 9
 <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
 	<vars>x,y,z</vars>
 	<basis>
 	<ncpoly>y*z-1</ncpoly>
 	<ncpoly>z*y-1</ncpoly>
 	<ncpoly>x*x-1</ncpoly>
 	<ncpoly>((x*y)^9)-1</ncpoly>
 	</basis>
 	<Comment>Hecke_group_9</Comment>
 </FREEALGEBRA>
Hecke group 10
 <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
 	<vars>x,y,z</vars>
 	<basis>
 	<ncpoly>y*z-1</ncpoly>
 	<ncpoly>z*y-1</ncpoly>
 	<ncpoly>x*x-1</ncpoly>
 	<ncpoly>((x*y)^10)-1</ncpoly>
 	</basis>
 	<Comment>Hecke_group_10</Comment>
 </FREEALGEBRA>
Hecke group 11
 <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
 	<vars>x,y,z</vars>
 	<basis>
 	<ncpoly>y*z-1</ncpoly>
 	<ncpoly>z*y-1</ncpoly>
 	<ncpoly>x*x-1</ncpoly>
 	<ncpoly>((x*y)^11)-1</ncpoly>
 	</basis>
 	<Comment>Hecke_group_11</Comment>
 </FREEALGEBRA>
Hecke group 12
 <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
 	<vars>x,y,z</vars>
 	<basis>
 	<ncpoly>y*z-1</ncpoly>
 	<ncpoly>z*y-1</ncpoly>
 	<ncpoly>x*x-1</ncpoly>
 	<ncpoly>((x*y)^12)-1</ncpoly>
 	</basis>
 	<Comment>Hecke_group_12</Comment>
 </FREEALGEBRA>
Hecke group 13
 <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
 	<vars>x,y,z</vars>
 	<basis>
 	<ncpoly>y*z-1</ncpoly>
 	<ncpoly>z*y-1</ncpoly>
 	<ncpoly>x*x-1</ncpoly>
 	<ncpoly>((x*y)^13)-1</ncpoly>
 	</basis>
 	<Comment>Hecke_group_13</Comment>
 </FREEALGEBRA>
Hecke group 14
 <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
 	<vars>x,y,z</vars>
 	<basis>
 	<ncpoly>y*z-1</ncpoly>
 	<ncpoly>z*y-1</ncpoly>
 	<ncpoly>x*x-1</ncpoly>
 	<ncpoly>((x*y)^14)-1</ncpoly>
 	</basis>
 	<Comment>Hecke_group_14</Comment>
 </FREEALGEBRA>
Hecke group 15
 <FREEALGEBRA createdAt="2015-03-02" createdBy="strohmeier">
 	<vars>x,y,z</vars>
 	<basis>
 	<ncpoly>y*z-1</ncpoly>
 	<ncpoly>z*y-1</ncpoly>
 	<ncpoly>x*x-1</ncpoly>
 	<ncpoly>((x*y)^15)-1</ncpoly>
 	</basis>
 	<Comment>Hecke_group_15</Comment>
 </FREEALGEBRA>
Hecke group 16
 <FREEALGEBRA createdAt="2016-03-02" createdBy="strohmeier">
 	<vars>x,y,z</vars>
 	<basis>
 	<ncpoly>y*z-1</ncpoly>
 	<ncpoly>z*y-1</ncpoly>
 	<ncpoly>x*x-1</ncpoly>
 	<ncpoly>((x*y)^16)-1</ncpoly>
 	</basis>
 	<Comment>Hecke_group_16</Comment>
 </FREEALGEBRA>
Hecke group 17
 <FREEALGEBRA createdAt="2017-03-02" createdBy="strohmeier">
 	<vars>x,y,z</vars>
 	<basis>
 	<ncpoly>y*z-1</ncpoly>
 	<ncpoly>z*y-1</ncpoly>
 	<ncpoly>x*x-1</ncpoly>
 	<ncpoly>((x*y)^17)-1</ncpoly>
 	</basis>
 	<Comment>Hecke_group_17</Comment>
 </FREEALGEBRA>