# Difference between revisions of "ApCoCoA-1:Heisenberg groups"

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

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

## Revision as of 13:09, 20 August 2013

#### Description

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

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);