ApCoCoA-1: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>