Documentation

Hyperbolic Lattices in MAGMA

The algorithm described in the paper Automorphism Groups of Hyperbolic Lattices is implemented in the MAGMA-package AutHyp.m which can be downloaded here. To use it in MAGMA, either include its path into MAGMA's spec-file (then it will be loaded in every MAGMA session) or type

   > Attach("/path/of/AutHyp.m");

during a current MAGMA session (then the package will only be attached for this current session).

A hyperbolic lattice is to be represented by its Gram matrix which we will call A from now on. Take for example

  
> A:=SymmetricMatrix([2,3,3,1,1,4]);
> A;
[2 3 1]
[3 3 1]
[1 1 4]
> pp:=GetPerfectPoints(A);
Got first perfect point. It has  9  D-minimal vectors
Found  4  directions of first point
Testing number  1
2 : Found point with  3 directions
Testing number  2
3 : Found point with  4 directions
4 : Found point with  3 directions
Testing number  3
Testing number  4

The intrinsic GetPerfectPoints returns a record which contains the following fields:

a list of cone test vectors (cf. equations (2) and (3) in the paper)

  
> pp`ConeTestVectors;
[
    (  0 1/2 1/6),
    (-3  2  0)
]

a transversal of the D-perfect points modulo the reduced automorphism group (represented as primitive integer vectors)
```
  
    > pp`Points;
[
    (-1  1  0),
    (-3  3  1),
    (-4  3  2),
    (-9 11  3)
]
  
 
```

a list of the lattices L(x) represented by their Gram matrices (cf. Lemma 3.12 in the paper)

  
> pp`Lat;
[
    [3 1]
    [1 4],

    [ 4 -1]
    [-1 14],

    [ 1  0]
    [ 0 11],

    [  2  -1]
    [ -1 116]
]

a list where each entry is a list of directions for each D-perfect point

  
> pp`directions;
[   
    [   
        (-1  1  1),
        (-1  0  0),
        (-1  1 -1),
        (-1  2  0)
    ],
    [   
        (-5  3  3),
        (-3  5  1),
        ( 1 -1 -1)
    ],
    [   
        (-29  21  15),
        (-7  3  3),
        (-17  15   9),
        ( 5 -3 -3)
    ],
    [   
        (-1  3 -1),
        ( 3 -5 -1),
        (-11  13   5)
    ]
]

a list of pairs representing the contiguity relations among the perfect points

  
> pp`Neighbours;                                                                                                                                                                             
[ <1, 2>, <2, 3>, <2, 4> ]

a list of the stabilizers of the D-perfect points

  
> pp`Stabilizers;
[
    MatrixGroup(3, Integer Ring)
    Generators:
        [ 1 -2  0]
        [ 0 -1  0]
        [ 0  0 -1],

    MatrixGroup(3, Integer Ring) of order 1,

    MatrixGroup(3, Integer Ring)
    Generators:
        [ 23 -18 -12]
        [ -8   5   4]
        [ 56 -42 -29]

        [ 23 -18 -12]
        [  0   1   0]
        [ 44 -36 -23],

    MatrixGroup(3, Integer Ring)
    Generators:
        [ 5 -8  0]
        [ 3 -5  0]
        [ 1 -2  1]
]

a list of tuples (i,j positive integers, v a vector and C a matrix) denoting that in the residue class graph the ith and jth perfect points are adjacent via the connecting element C. The actual neighbour of point number i is the vector v.

  
> pp`Connectors;
[ 
    <
        1,
  
        1,

        (-2  1  0),

        [ 5 -2  0]
        [ 3 -1  0]
        [ 1  0 -1]
    >,

    <
        2,

        1,

        (-1  1  0),

        [1 0 0]
        [0 1 0]
        [0 0 1]
    >,

    <
        3,

        2,

        (-37  27  19),

        [ 23 -18 -12]
        [ -8   5   4]
        [ 56 -42 -29]
    >,
  
    <
        4,
  
        4,
  
        (-31  37  13),
  
        [ 49 -60 -20]
        [ 25 -31 -10]
        [ 45 -54 -19]
    >,
  
    <
        4,
  
        2,
  
        (-5  7  1),
  
        [ 5 -8  0]
        [ 3 -5  0]
        [ 1 -2  1]
    >
]

a list of matrices which contains of the generators of the stabilizers and the connecting elements. These matrices generate the reduced automorphism group of A.

  
> pp`AutGrp;       
[
    [ 1 -2  0]
    [ 0 -1  0]
    [ 0  0 -1],

    [ 23 -18 -12]
    [ -8   5   4]
    [ 56 -42 -29],

    [ 23 -18 -12]
    [  0   1   0]
    [ 44 -36 -23],

    [ 5 -8  0]
    [ 3 -5  0]
    [ 1 -2  1],

    [ 5 -2  0]
    [ 3 -1  0]
    [ 1  0 -1],

    [ 23 -18 -12]
    [ -8   5   4]
    [ 56 -42 -29],

    [ 49 -60 -20]
    [ 25 -31 -10]
    [ 45 -54 -19],

    [ 5 -8  0]
    [ 3 -5  0]
    [ 1 -2  1]
]

a boolean value that indicates whether the algorithm was executed completely
```
  
> pp`Complete;   
true
  
```

The intrinsic AutGroup returns the actual automorphism group as a finitely generated matrix group and the number of inequivalent D-perfect points. By default, this intrinsic uses the Watson process, thus to be more precise in this case the second return value is the number of inequivalent D-perfect points of the Watson matrix of A. By the option Watson:=false the Watson process is not used anymore, but in average this results in a longer computation time.

 
   
> AutGroup(A);                                   
Got first perfect point. It has  9  D-minimal vectors
Found  4  directions of first point
Testing number  1
2 : Found point with  3 directions
Testing number  2
3 : Found point with  4 directions
4 : Found point with  3 directions
Testing number  3
Testing number  4
Starting stabilizer computation.
    [ 1 -2  0]
    [ 0 -1  0]
    [ 0  0 -1]

    [ 23 -18 -12]
    [ -8   5   4]
    [ 56 -42 -29]

    [ 23 -18 -12]
    [  0   1   0]
    [ 44 -36 -23]

    [ 5 -8  0]
    [ 3 -5  0]
    [ 1 -2  1]

    [ 5 -2  0]
    [ 3 -1  0]
    [ 1  0 -1]

    [ 49 -60 -20]
    [ 25 -31 -10]
    [ 45 -54 -19]

    [-1  0  0]
    [ 0 -1  0]
    [ 0  0 -1]
4

Hyperbolic Lattices in MAGMA

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