ApCoCoA-1:Heisenberg groups

Heisenberg groups

Description

The Heisenberg group is the group of 3x3 upper triangular matrices of the form

1 a c

0 1 b

0 0 1

Heisenberg groups are often used in quantum mechanics and also occurs in fourier analysis. A representation is given by:

 H(2k+1) = <a_{1},...,a_{k},b_{1},...,b_{k},c | [a_{i},b_{i}] = c, [a_{i},c] = [b_{i},c], [a_{i},b_{j}] = 1 for all i != j

(Reference: Ernst Binz & Sonja Pods (2008) Geometry of Heisenberg Groups American Mathematical Society)

Computation

 /*Use the ApCoCoA package ncpoly.*/
 
 // Number of Heisenberg group
 MEMORY.N:=1;
 
 // a invers to d and b invers to e and c invers to f
 Use ZZ/(2)[a[1..MEMORY.N],b[1..MEMORY.N],c,d[1..MEMORY.N],e[1..MEMORY.N],f];
 NC.SetOrdering("LLEX");
 Define CreateRelationsHeisenberg()
   Relations:=[];
   // add the relations of the invers elements ad = da = be = eb = cf = fc = 1
   Append(Relations,[[c,f],[1]]);
   Append(Relations,[[f,c],[1]]);
   For Index1 := 1 To MEMORY.N Do
     Append(Relations,[[a[Index1],d[Index1]],[1]]);
     Append(Relations,[[d[Index1],a[Index1]],[1]]);
     Append(Relations,[[b[Index1],e[Index1]],[1]]);
     Append(Relations,[[e[Index1],b[Index1]],[1]]);
   EndFor;
   
   // add the relation [a_{i}, b_{i}] = c
   For Index2 := 1 To MEMORY.N Do
     Append(Relations,[[a[Index2],b[Index2],d[Index2],e[Index2]],[c]]);
   EndFor;
   
   // add the relation [a_{i}, c] = [b_i, c]
   For Index3 := 1 To MEMORY.N Do
     Append(Relations,[[a[Index3],c,d[Index3],f],[b[Index3],c,e[Index3],f]]);
   EndFor;
   
   // add the relation [a_{i}, b_{j}] = 1 for all i != j
   For Index4 := 1 To MEMORY.N Do
     For Index5 := 1 To MEMORY.N Do
       If Index4 <> Index5 Then
         Append(Relations,[[a[Index4],b[Index5],d[Index4],e[Index5]],[1]]);
       EndIf;
     Endfor;
   EndFor;
   
   Return Relations;
 EndDefine;
 
 Relations:=CreateRelationsHeisenberg();
 Relations;
 Size(Relations);
 GB:=NC.GB(Relations,31,1,100,1000);
 Size(GB);