Lattices with isometry

We call lattice with isometry any pair $(L, f)$ consisting of an integer lattice $L$ together with an isometry $f \in O(L)$. We refer to the section about Integer Lattices of the documentation for new users.

In Oscar, such a pair is encoded in the type called ZZLatWithIsom:

ZZLatWithIsom — Type

ZZLatWithIsom

A container type for pairs $(L, f)$ consisting of an integer lattice $L$ of type ZZLat and an isometry $f$ given as a QQMatrix representing the action on the basis matrix of $L$.

We store the ambient space $V$ of $L$ together with an isometry $f_a$ inducing $f$ on $L$ seen as a pair $(V, f_a)$ of type QuadSpaceWithIsom. We moreover store the order $n$ of $f$, which can be finite or infinite.

To construct an object of type ZZLatWithIsom, see the following examples:

Examples

One first way to construct such object, is by entering directly the lattice with an isometry. The isometry can be a honnest isometry of the lattice, or it can be an isometry of the ambient space preserving the lattice. Depending on this choice, one should enter the appropriate boolean value ambient_representation. This direct construction is done through the constructors integer_lattice_with_isometry.

julia> L = root_lattice(:E, 6);

julia> f = matrix(QQ, 6, 6, [ 1  2  3  2  1  1;
                             -1 -2 -2 -2 -1 -1;                                                
                              0  1  0  0  0  0;      
                              1  0  0  0  0  0;
                             -1 -1 -1  0  0 -1;
                              0  0  1  1  0  1]);

julia> Lf = integer_lattice_with_isometry(L, f; ambient_representation=false)
Integer lattice of rank 6 and degree 6
  with isometry of finite order 8
  given by
  [ 1    2    3    2    1    1]
  [-1   -2   -2   -2   -1   -1]
  [ 0    1    0    0    0    0]
  [ 1    0    0    0    0    0]
  [-1   -1   -1    0    0   -1]
  [ 0    0    1    1    0    1]

julia> B = matrix(QQ,1,6, [1   2   3   1   -1   3]);

julia> I = lattice_in_same_ambient_space(L, B); # This is the invariant sublattice L^f

julia> If = integer_lattice_with_isometry(I, ambient_isometry(Lf))
Integer lattice of rank 1 and degree 6
  with isometry of finite order 1
  given by
  [1]

julia> integer_lattice_with_isometry(I; neg=true)
Integer lattice of rank 1 and degree 6
  with isometry of finite order 2
  given by
  [-1]

Another way to construct such objects is to see them as sub-objects of their ambient space, of type QuadSpaceWithIsom. Through the constructors lattice(::QuadSpaceWithIsom) and lattice_in_same_ambient_space(::ZZLatWithIsom, ::MatElem), one can then construct lattices with isometry for free, in a given space, as long as the module they define is preserved by the fixed isometry of the ambient space.

Examples

julia> G = matrix(QQ, 6, 6 , [ 3 1 -1 1 0 0;
                               1 3  1 1 1 1;
                              -1 1  3 0 0 1;
                               1 1  0 4 2 2;
                               0 1  0 2 4 2;
                               0 1  1 2 2 4]);

julia> V = quadratic_space(QQ, G);

julia> f = matrix(QQ, 6, 6, [ 1 0  0 0 0  0
                              0 0 -1 0 0  0
                             -1 1 -1 0 0  0
                              0 0  0 1 0 -1
                              0 0  0 0 0 -1
                              0 0  0 0 1 -1]);

julia> Vf = quadratic_space_with_isometry(V, f);

julia> Lf = lattice(Vf)
Integer lattice of rank 6 and degree 6
  with isometry of finite order 3
  given by
  [ 1   0    0   0   0    0]
  [ 0   0   -1   0   0    0]
  [-1   1   -1   0   0    0]
  [ 0   0    0   1   0   -1]
  [ 0   0    0   0   0   -1]
  [ 0   0    0   0   1   -1]

julia> B = matrix(QQ, 4, 6, [1 0 3 0 0 0;
                             0 1 1 0 0 0;
                             0 0 0 0 1 0;
                             0 0 0 0 0 1]);

julia> Cf = lattice(Vf, B)  # coinvariant sublattice L_f
Integer lattice of rank 4 and degree 6
  with isometry of finite order 3
  given by
  [-2   3   0    0]
  [-1   1   0    0]
  [ 0   0   0   -1]
  [ 0   0   1   -1]

julia> Cf2 = lattice_in_same_ambient_space(Lf, B)
Integer lattice of rank 4 and degree 6
  with isometry of finite order 3
  given by
  [-2   3   0    0]
  [-1   1   0    0]
  [ 0   0   0   -1]
  [ 0   0   1   -1]

julia> Cf == Cf2
true

The last equality of the last example shows why we care about ambient context: the two pairs of lattice with isometry Cf and Cf2 are basically the same mathematical objects. Indeed, they lie in the same space, defines the same module and their respective isometries are induced by the same isometry of the ambient space. As for regular ZZLat, as soon as the lattices are in the same ambient space, we can compare them as $\mathbb Z$-modules, endowed with an isometry.

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.