Methods for group actions on ZZLat

[StevellM]

Introduce new features for (finite) groups of isometries of ZZLat.

List of new features:

• Extra sanity checks via the new is_*_isometry_* methods;
• Convenient methods to change matrix representation of isometries of lattices;
• Constructors for the stable orthogonal group and special orthogonal group of definite lattices;
• Constructors for the stable/special subgroup of a finite group of isometries;
• Methods for computing:
  • joint stabilizers of a finite set of vectors in the orthogonal group of a lattice;
  • pointwize stabilizer of a lattice with definite complement;
  • stabilizer of a definite sublattice with definite complement;
• Saturation of group of isometries with definite coinvariant sublattice.

List of changes:

• Allow invariant_coinvariant_pair(::ZZLat, ...) to take as input a single matrix, or a list of matrices;
• In discriminant_representation, give the option not to compute the full codomain (i.e. orthogonal group of the discriminant group) but just the image of the map (via the new keyword argument full).
• Add the possibility to give a list of FinGenAbGroupHom or TorQuadModuleMap in the internal function _orthogonal_group(::TorQuadModule, ::Vector) (previously the second output only allowed for ZZMatrix)

To be discussed:

• The "stabilizer" methods currently assume that all lattices are even (to use discriminant group). They could technically be upgraded for all integral lattices (and also, to some extent, to all lattices).
• In lines 410 and 665 of the file finite_group_actions.jl, we use stabilizer(..., on_lattices) which can be sometimes slow. One way to bypass that would be, in the integral case, to look at the stabilizer of some discriminant subgroups or of some gluing. It is not clear yet which option is the best.

To be done:

• Add tests for the new features.
• ~ Discuss whether we extend the "stabilizer" functionalities to all (integral) lattices.~
• Discuss whether we replace the on_lattices calls by stabilizers of discriminant subgroups and gluings.
• Related to the previous point: discuss whether we want to add general functionalities related to stabilizers of discriminant subgroups and gluings.

@simonbrandhorst

[simonbrandhorst]

There is a lot of copy pasting in the documentation which makes maintainance more difficult. Maybe you can consolidate it a bit?
What julia sometimes does is: "See some_function for the arguments".

[simonbrandhorst]

We should not insist on finiteness. I have usecases where OS is infinite (coming from Borcherds method)

[simonbrandhorst]

Best define precisely the action and what is stabilized precisely here.

[simonbrandhorst]

Suggested change

	stabilizer_sublattice_in_orthogonal_group(
	setwise_stabilizer_in_orthogonal_group(

?

[StevellM]

Good suggestion, thank you!

[simonbrandhorst]

Suggested change

	P, _ = stabilizer(P, S, on_lattices)
	P, _ = stabilizer(P, S, on_lattices)

Perhaps it would be faster to go via the discriminant groups?
How do these compare in terms of speed?

[simonbrandhorst]

Here and in other places did you try the following style? At least the headline #Arguments should be there I think.:

Suggested change

	Given a lattice ``L`` and a finite group of isometries ``G`` of ``L`` with
	definite coinvariant sublattice``S`` , return the saturation of ``G`` in the
	orthogonal group ``O(L)`` of ``L``, that is the pointwize stabilizer in ``O(L)``
	of its invariant sublattice.
	If `ambient_representation` is set to `true`, the group ``G`` is considered as
	a group of $\mathbb{Q}$-linear automorphisms on the ambient space of ``L``.
	If `check` is set to `true`, the function tests whether the group ``G`` is
	finite, whether it defines a group of isometries of the lattice ``L`` and
	whether the coinvariant lattice ``S`` is definite.
	If `special` is set to `true`, the function returns the saturation of ``G``
	in the special orthogonal group ``SO(L)`` of ``L``. Note that this requires
	``G`` to consist also of special isometries. If `check` is set to `true`, the
	function tests whether all isometries in ``G`` have determinant `.
	If `stable` is set to `true`, the function returns the saturation of ``G``
	in the stable orthogonal group ``O^\#(L)`` of ``L``. Note that this requires
	``G`` to consist also of stable isometries. If `check` is set to `true`, the
	function tests whether all isometries in ``G`` act trivially on the
	discriminant group of ``L``.
	The keyword arguments in `kwargs` are optional arguments for the computation
	of isometry group of definite lattices (see [`isometry_group(::ZZLat)`](@ref)).
	Given a lattice ``L`` and a finite group of isometries ``G`` of ``L`` with
	definite coinvariant sublattice``S`` , return the saturation of ``G`` in the
	orthogonal group ``O(L)`` of ``L``, that is the pointwize stabilizer in ``O(L)``
	of its invariant sublattice.
	# Arguments
	- If `ambient_representation` is set to `true`, the group ``G`` is considered as
	a group of $\mathbb{Q}$-linear automorphisms on the ambient space of ``L``.
	- If `check` is set to `true`, the function tests whether the group ``G`` is
	finite, whether it defines a group of isometries of the lattice ``L`` and
	whether the coinvariant lattice ``S`` is definite.
	- If `special` is set to `true`, the function returns the saturation of ``G``
	in the special orthogonal group ``SO(L)`` of ``L``. Note that this requires
	``G`` to consist also of special isometries. If `check` is set to `true`, the
	function tests whether all isometries in ``G`` have determinant `.
	- If `stable` is set to `true`, the function returns the saturation of ``G``
	in the stable orthogonal group ``O^\#(L)`` of ``L``. Note that this requires
	``G`` to consist also of stable isometries. If `check` is set to `true`, the
	function tests whether all isometries in ``G`` act trivially on the
	discriminant group of ``L``.
	- The keyword arguments in `kwargs` are optional arguments for the computation
	of isometry group of definite lattices (see [`isometry_group(::ZZLat)`](@ref)).