ApCoCoA:Coxeter Group F4

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Description

The F4 group is a Coxeter group. Their relations results of a Matrix, the Coxetermatrix. The Matrix with i lines and j columns gives the following relations:

<r_1,...,r_n|(r_ir_j)^m_ij

-the relation mii means: (r_ir_i)^1=1 for all i

-and other generators r_i, r_j commute.

F4 has the following presentation:

F4 = <v,x,y,z | v^2 = x^2 = y^2 = z^2 = (vx)^3 =(vy)^2 = (vz)^2 =(xy)^4 = (xz)^2 =(yz)^3 = 1>

Reference

not found yet

Computation

/*Use the ApCoCoA package ncpoly.*/
 
// Define Coxeter matrix
F:=Mat([[1,3,2,2],[3,1,4,2],[2,4,1,3],[2,2,3,1]]);
 
MEMORY.F1 := F[1,2]; //=F[2,1]
MEMORY.F2 := F[1,3]; //=F[3,1]
MEMORY.F3 := F[1,4]; //=F[4,1]
MEMORY.F4 := F[2,3]; //=F[3,2]
MEMORY.F5 := F[2,4]; //=F[4,2]
MEMORY.F6 := F[3,4]; //=F[4,3]


Use ZZ/(2)[v,x,y,z];
NC.SetOrdering("LLEX");

Define CreateRelationsCoxetergroupF4()
 Relations:=[];
   
 //add the inverse relations
 Append(Relations,[[v,v],[1]]);
 Append(Relations,[[x,x],[1]]);
 Append(Relations,[[y,y],[1]]);
 Append(Relations,[[z,z],[1]]);
 
   // add the relation (vx)^F[1,2] = 1
  Buffer12:=[];
  For Index1 := 1 To MEMORY.F1 Do
  	Append(Buffer12,v);
  	Append(Buffer12,x);
  EndFor;
  Append(Relations,[Buffer12,[1]]);
  
  // add the relation (xv)^F[2,1] = 1
  Buffer21:=[];
  For Index1 := 1 To MEMORY.F1 Do
  	Append(Buffer21,x);
  	Append(Buffer21,v);
  EndFor;
  Append(Relations,[Buffer21,[1]]); 

   // add the relation (vy)^F[1,3] = 1
  Buffer13:=[];
  For Index1 := 1 To MEMORY.F2 Do
  	Append(Buffer13,v);
  	Append(Buffer13,y);
  EndFor;
  Append(Relations,[Buffer13,[1]]);
  
  // add the relation (yv)^F[3,1] = 1
  Buffer31:=[];
  For Index1 := 1 To MEMORY.F2 Do
  	Append(Buffer31,v);
  	Append(Buffer31,y);
  EndFor;
  Append(Relations,[Buffer31,[1]]); 

   // add the relation (vz)^F[1,4] = 1
  Buffer14:=[];
  For Index1 := 1 To MEMORY.F3 Do
  	Append(Buffer14,v);
  	Append(Buffer14,z);
  EndFor;
  Append(Relations,[Buffer14,[1]]);
  
  // add the relation (zv)^F[4,1] = 1
  Buffer41:=[];
  For Index1 := 1 To MEMORY.F3 Do
  	Append(Buffer41,z);
  	Append(Buffer41,v);
  EndFor;
  Append(Relations,[Buffer41,[1]]); 
 
     // add the relation (xy)^F[2,3] = 1
  Buffer23:=[];
  For Index1 := 1 To MEMORY.F4 Do
  	Append(Buffer23,x);
  	Append(Buffer23,y);
  EndFor;
  Append(Relations,[Buffer23,[1]]);
  
  // add the relation (yx)^F[3,2] = 1
  Buffer32:=[];
  For Index1 := 1 To MEMORY.F4 Do
  	Append(Buffer32,y);
  	Append(Buffer32,x);
  EndFor;
  Append(Relations,[Buffer32,[1]]); 

     // add the relation (xz)^F[2,4] = 1
  Buffer24:=[];
  For Index1 := 1 To MEMORY.F5 Do
  	Append(Buffer24,x);
  	Append(Buffer24,z);
  EndFor;
  Append(Relations,[Buffer24,[1]]);
  
  // add the relation (yx)^F[4,2] = 1
  Buffer42:=[];
  For Index1 := 1 To MEMORY.F5 Do
  	Append(Buffer42,z);
  	Append(Buffer42,x);
  EndFor;
  Append(Relations,[Buffer42,[1]]); 

     // add the relation (yz)^F[3,4] = 1
  Buffer34:=[];
  For Index1 := 1 To MEMORY.F6 Do
  	Append(Buffer34,y);
  	Append(Buffer34,z);
  EndFor;
  Append(Relations,[Buffer34,[1]]);
  
  // add the relation (zy)^F[4,3] = 1
  Buffer43:=[];
  For Index1 := 1 To MEMORY.F6 Do
  	Append(Buffer43,z);
  	Append(Buffer43,y);
  EndFor;
  Append(Relations,[Buffer43,[1]]); 


  Return Relations;
EndDefine;

Relations:=CreateRelationsCoxetergroupF4();
Relations;

Gb:=NC.GB(Relations,31,1,100,1000);
Gb;

Example in Symbolic Data Format

<FREEALGEBRA createdAt="2014-07-30" createdBy="strohmeier">
	<vars>v,x,y,z</vars>
	<basis>
	<ncpoly>v*v-1</ncpoly>		
	<ncpoly>x*x-1</ncpoly>
	<ncpoly>y*y-1</ncpoly>
	<ncpoly>z*z-1</ncpoly>
	<ncpoly>(v*x)^3-1</ncpoly>
	<ncpoly>(x*v)^3-1</ncpoly>
	<ncpoly>(v*y)^2-1</ncpoly>
	<ncpoly>(y*v)^2-1</ncpoly>
	<ncpoly>(v*z)^2-1</ncpoly>
	<ncpoly>(z*v)^2-1</ncpoly>
	<ncpoly>(x*y)^4-1</ncpoly>
	<ncpoly>(y*x)^4-1</ncpoly>
	<ncpoly>(x*z)^2-1</ncpoly>
	<ncpoly>(z*x)^2-1</ncpoly>
	<ncpoly>(y*z)^3-1</ncpoly>
	<ncpoly>(z*y)^3-1</ncpoly>
	</basis>
	<Comment>Coxeter_Group_F4</Comment>
</FREEALGEBRA>