ApCoCoA-1: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>