Difference between revisions of "ApCoCoA-1:Thompson group"
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− | === <div id="Thompson_groups">[[:ApCoCoA:Symbolic data#Thompson_group|Thompson | + | === <div id="Thompson_groups">[[:ApCoCoA:Symbolic data#Thompson_group|Thompson Group]]</div> === |
==== Description ==== | ==== Description ==== | ||
The Thompson group can be regarded as the group of piecewise-linear, orientation-preserving homeomorphisms of the unit interval which have breakpoints only at dyadic points and on intervals of differentiability the slopes are powers of | The Thompson group can be regarded as the group of piecewise-linear, orientation-preserving homeomorphisms of the unit interval which have breakpoints only at dyadic points and on intervals of differentiability the slopes are powers of | ||
two. A representation is given by: | two. A representation is given by: | ||
T = <a,b | [ab^{-1},a^{-1}ba] = [ab^{-1},a^{-2}ba^{2}] = 1> | T = <a,b | [ab^{-1},a^{-1}ba] = [ab^{-1},a^{-2}ba^{2}] = 1> | ||
+ | or alternative: | ||
+ | Th = <x_{0},x_{1},x_{2},... | x_{k}^{-1}x_{n}x_{k} = x_{n+1} for all k < n> | ||
==== Reference ==== | ==== Reference ==== | ||
Line 9: | Line 11: | ||
==== Computation ==== | ==== Computation ==== | ||
− | /*Use the ApCoCoA package ncpoly.*/ | + | /*Use the ApCoCoA package ncpoly.*/ |
Use ZZ/(2)[a,b,c,d]; | Use ZZ/(2)[a,b,c,d]; | ||
Line 28: | Line 30: | ||
//add the relation [ad,a^{-1}ba] = 1 | //add the relation [ad,a^{-1}ba] = 1 | ||
− | // the commutator of [ad,a^{-2}ba^2] is a,d,c,c,b,a,a,b,c,c | + | // the commutator of [ad,a^{-2}ba^2] is a,d,c,c,b,a,a,b,c,c^2,d,a^2 |
− | Append(Relations,[[a,d,c,c,b,a,a,b,c,c | + | Append(Relations,[[a,d,c,c,b,a,a,b,c,c^2,d,a^2],[1]]); |
Return Relations; | Return Relations; | ||
Line 62: | Line 64: | ||
[ad,ccbaa]=a*d*c*c*b*a*a*b*c*c^2*d*a^2 | [ad,ccbaa]=a*d*c*c*b*a*a*b*c*c^2*d*a^2 | ||
</Comment> | </Comment> | ||
+ | |||
+ | ==== Alternative Computation ==== | ||
+ | |||
+ | /*Use the ApCoCoA package ncpoly.*/ | ||
+ | |||
+ | // Define the variable k,n of the thompson group | ||
+ | MEMORY.N:=5; | ||
+ | |||
+ | Use ZZ/(2)[x[1..MEMORY.N],y[1..MEMORY.N]]; | ||
+ | NC.SetOrdering("LLEX"); | ||
+ | |||
+ | Define CreateRelationsthomp() | ||
+ | Relations:=[]; | ||
+ | For Index1 := 1 To MEMORY.N Do | ||
+ | Append(Relations,[[x[Index1],y[Index1]],[1]]); | ||
+ | Append(Relations,[[y[Index1],x[Index1]],[1]]); | ||
+ | EndFor; | ||
+ | |||
+ | For Index1 := 2 To MEMORY.N-1 Do | ||
+ | For Index2 := 1 To MEMORY.N-2 Do | ||
+ | If (Index1 > Index2) Then | ||
+ | Append(Relations,[[y[Index2],x[Index1],x[Index2]],[x[Index1+1]]]); | ||
+ | EndIf | ||
+ | EndFor; | ||
+ | EndFor; | ||
+ | Return Relations; | ||
+ | EndDefine; | ||
+ | |||
+ | Relations:=CreateRelationsthomp(); | ||
+ | Relations; | ||
+ | |||
+ | Gb:=NC.GB(Relations,31,1,100,1000); | ||
+ | Gb; | ||
+ | |||
+ | ====Example in Symbolic Data Format==== | ||
+ | <FREEALGEBRA createdAt="2014-03-28" createdBy="strohmeier"> | ||
+ | <vars>x1,x2,x3,x4,x5,y1,y2,y3,y4,y5</vars> | ||
+ | <uptoDeg>4</uptoDeg> | ||
+ | <basis> | ||
+ | <ncpoly>x1*y1-1</ncpoly> | ||
+ | <ncpoly>y1*x1-1</ncpoly> | ||
+ | <ncpoly>x2*y2-1</ncpoly> | ||
+ | <ncpoly>y2*x2-1</ncpoly> | ||
+ | <ncpoly>x3*y3-1</ncpoly> | ||
+ | <ncpoly>y3*x3-1</ncpoly> | ||
+ | <ncpoly>x4*y4-1</ncpoly> | ||
+ | <ncpoly>y4*x4-1</ncpoly> | ||
+ | <ncpoly>x5*y5-1</ncpoly> | ||
+ | <ncpoly>y5*x5-1</ncpoly> | ||
+ | <ncpoly>y1*x2*x1-x3</ncpoly> | ||
+ | <ncpoly>y1*x3*x1-x4</ncpoly> | ||
+ | <ncpoly>y1*x4*x1-x5</ncpoly> | ||
+ | <ncpoly>y2*x3*x2-x4</ncpoly> | ||
+ | <ncpoly>y2*x4*x2-x5</ncpoly> | ||
+ | <ncpoly>y3*x4*x3-x5</ncpoly> | ||
+ | </basis> | ||
+ | <Comment>The partial LLex Gb has 126 elements</Comment> | ||
+ | <Comment>Thompson_group_alt5</Comment> | ||
+ | </FREEALGEBRA> |
Latest revision as of 21:05, 22 April 2014
Description
The Thompson group can be regarded as the group of piecewise-linear, orientation-preserving homeomorphisms of the unit interval which have breakpoints only at dyadic points and on intervals of differentiability the slopes are powers of two. A representation is given by:
T = <a,b | [ab^{-1},a^{-1}ba] = [ab^{-1},a^{-2}ba^{2}] = 1>
or alternative:
Th = <x_{0},x_{1},x_{2},... | x_{k}^{-1}x_{n}x_{k} = x_{n+1} for all k < n>
Reference
NEW PRESENTATIONS OF THOMPSON'S GROUPS AND APPLICATIONS: UFFE HAAGERUP AND GABRIEL PICIOROAGA
Computation
/*Use the ApCoCoA package ncpoly.*/ Use ZZ/(2)[a,b,c,d]; NC.SetOrdering("LLEX"); Define CreateRelationsThompson() Relations:=[]; // add the inverse relations Append(Relations,[[a,c],[1]]); Append(Relations,[[c,a],[1]]); Append(Relations,[[b,d],[1]]); Append(Relations,[[d,b],[1]]); //add the relation [ad,a^{-1}ba] = 1 // the commutator of [ad,a^{-1}ba] is a,d,c,b,a,b,c,c,d,a Append(Relations,[[a,d,c,b,a,b,c,c,d,a],[1]]); //add the relation [ad,a^{-1}ba] = 1 // the commutator of [ad,a^{-2}ba^2] is a,d,c,c,b,a,a,b,c,c^2,d,a^2 Append(Relations,[[a,d,c,c,b,a,a,b,c,c^2,d,a^2],[1]]); Return Relations; EndDefine; Relations:=CreateRelationsThompson(); Relations; Gb:=NC.GB(Relations,31,1,100,1000); Gb;
Example in Symbolic Data Format
<FREEALGEBRA createdAt="2014-01-20" createdBy="strohmeier"> <vars>a,b,c,d</vars> <uptoDeg>11</uptoDeg> <basis> <ncpoly>a*c-1</ncpoly> <ncpoly>c*a-1</ncpoly> <ncpoly>b*d-1</ncpoly> <ncpoly>d*b-1</ncpoly> <ncpoly>a*d*c*b*a*b*c*c*d*a-1</ncpoly> <ncpoly>a*d*c*c*b*a*a*b*c*c^2*d*a^2-1</ncpoly> </basis> <Comment>The partial LLex Gb has 393 elements</Comment> <Comment>Thompson_group</Comment> </FREEALGEBRA> <Comment> Commutators [g,h] = ghg^{-1}h^{-1} [ad,cba]=a*d*c*b*a*b*c*c*d*a [ad,ccbaa]=a*d*c*c*b*a*a*b*c*c^2*d*a^2 </Comment>
Alternative Computation
/*Use the ApCoCoA package ncpoly.*/ // Define the variable k,n of the thompson group MEMORY.N:=5; Use ZZ/(2)[x[1..MEMORY.N],y[1..MEMORY.N]]; NC.SetOrdering("LLEX"); Define CreateRelationsthomp() Relations:=[]; For Index1 := 1 To MEMORY.N Do Append(Relations,[[x[Index1],y[Index1]],[1]]); Append(Relations,[[y[Index1],x[Index1]],[1]]); EndFor; For Index1 := 2 To MEMORY.N-1 Do For Index2 := 1 To MEMORY.N-2 Do If (Index1 > Index2) Then Append(Relations,[[y[Index2],x[Index1],x[Index2]],[x[Index1+1]]]); EndIf EndFor; EndFor; Return Relations; EndDefine; Relations:=CreateRelationsthomp(); Relations; Gb:=NC.GB(Relations,31,1,100,1000); Gb;
Example in Symbolic Data Format
<FREEALGEBRA createdAt="2014-03-28" createdBy="strohmeier"> <vars>x1,x2,x3,x4,x5,y1,y2,y3,y4,y5</vars> <uptoDeg>4</uptoDeg> <basis> <ncpoly>x1*y1-1</ncpoly> <ncpoly>y1*x1-1</ncpoly> <ncpoly>x2*y2-1</ncpoly> <ncpoly>y2*x2-1</ncpoly> <ncpoly>x3*y3-1</ncpoly> <ncpoly>y3*x3-1</ncpoly> <ncpoly>x4*y4-1</ncpoly> <ncpoly>y4*x4-1</ncpoly> <ncpoly>x5*y5-1</ncpoly> <ncpoly>y5*x5-1</ncpoly> <ncpoly>y1*x2*x1-x3</ncpoly> <ncpoly>y1*x3*x1-x4</ncpoly> <ncpoly>y1*x4*x1-x5</ncpoly> <ncpoly>y2*x3*x2-x4</ncpoly> <ncpoly>y2*x4*x2-x5</ncpoly> <ncpoly>y3*x4*x3-x5</ncpoly> </basis> <Comment>The partial LLex Gb has 126 elements</Comment> <Comment>Thompson_group_alt5</Comment> </FREEALGEBRA>