# ApCoCoA-1:Coxeter groups

### Full Icosahedral Group

#### Description

The full icosahedral 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.

H3 has the following presentation:

```H3 = <x,y,z | x^2 = y^2 = z^2 = (xy)^2 = (xz)^2 =(yz)^2 = 1 >
```

#### Computation

```/*Use the ApCoCoA package ncpoly.*/

// Define Coxeter matrix of H3
H:=Mat([[1,5,2],[5,1,3],[2,3,1]]);

MEMORY.H1 := H[1,2]; //=H[2,1]
MEMORY.H2 := H[1,3]; //=H[3,1]
MEMORY.H3 := H[2,3]; //=H[3,2]

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

Define CreateRelationsCoxetergroupF4()
Relations:=[];

Append(Relations,[[x,x],]);
Append(Relations,[[y,y],]);
Append(Relations,[[z,z],]);

// add the relation (xy)^H[1,2] = 1
Buffer12:=[];
For Index1 := 1 To MEMORY.H1 Do
Append(Buffer12,x);
Append(Buffer12,y);
EndFor;
Append(Relations,[Buffer12,]);

// add the relation (yx)^H[2,1] = 1
Buffer21:=[];
For Index1 := 1 To MEMORY.H1 Do
Append(Buffer21,y);
Append(Buffer21,x);
EndFor;
Append(Relations,[Buffer21,]);

// add the relation (xz)^H[1,3] = 1
Buffer13:=[];
For Index1 := 1 To MEMORY.H2 Do
Append(Buffer13,x);
Append(Buffer13,z);
EndFor;
Append(Relations,[Buffer13,]);

// add the relation (zx)^H[3,1] = 1
Buffer31:=[];
For Index1 := 1 To MEMORY.H2 Do
Append(Buffer31,z);
Append(Buffer31,x);
EndFor;
Append(Relations,[Buffer31,]);

// add the relation (yz)^H[2,3] = 1
Buffer23:=[];
For Index1 := 1 To MEMORY.H3 Do
Append(Buffer23,y);
Append(Buffer23,z);
EndFor;
Append(Relations,[Buffer23,]);

// add the relation (zy)^H[3,2] = 1
Buffer32:=[];
For Index1 := 1 To MEMORY.H3 Do
Append(Buffer32,z);
Append(Buffer32,y);
EndFor;
Append(Relations,[Buffer32,]);

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>x,y,z</vars>
<basis>
<ncpoly>x*x-1</ncpoly>
<ncpoly>y*y-1</ncpoly>
<ncpoly>z*z-1</ncpoly>
<ncpoly>(x*y)^5-1</ncpoly>
<ncpoly>(y*x)^5-1</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_H3</Comment>
</FREEALGEBRA>
```