#9504: Add support for toric sublattices
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Reporter: novoselt | Owner: mhampton
Type: enhancement | Status: needs_work
Priority: major | Milestone: sage-4.5.2
Component: geometry | Keywords:
Author: Andrey Novoseltsev | Upstream: N/A
Reviewer: | Merged:
Work_issues: quotient lattices |
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Changes (by novoselt):
* cc: vbraun (added)
* work_issues: documentation and quotients => quotient lattices
Comment:
Hi Volker,
The new version of the patch is completely documented, does not brake
anything in existing modules anymore, and in general, I think, provides
quite satisfactory support for toric sublattices, with the exception of
`dual` which requires quotient lattices and I need some input on them.
For sublattices, as before, I have followed `ZZ^n` behaviour with minimal
interference and it seems to work just fine. For quotients it is probably
not the way to go, since we are considering a special case of quotients
without torsion. Look at the following example:
{{{
sage: N = ToricLattice(3)
sage: Ns = N.submodule_with_basis([N(1,2,3)])
sage: Ns
Free module of degree 3 and rank 1 over Integer Ring
User basis matrix:
[1 2 3]
sage: Ns.basis()
[
N(1, 2, 3)
]
sage: Ns(N(1,2,3))
N(1, 2, 3)
sage: N.dual().gen(0) * Ns.gen(0)
1
sage: Q = N/Ns
sage: Q
Finitely generated module V/W over Integer Ring with invariants (0, 0)
sage: Q.gens()
((1, 0), (0, 1))
sage: Q.gen(0).lift()
N(0, 0, 1)
sage: Q(N(3,2,1))
(-8, -4)
sage: N.dual().gen(0) * Q.gen(0)
...
TypeError: unsupported operand parent(s) for '*':
'3-d lattice M' and 'Finitely generated module V/W
over Integer Ring with invariants (0, 0)'
}}}
Elements of `N` and its sublattice are actually represented by elements of
`N` (although elements of `Ns` know internally that they belong to `Ns`,
i.e. their `parent` is `Ns`). Elements of `Q`, on the other hand, are
represented by their coordinates in some basis. The natural dual of `Q` is
some sublattice of `N.dual()`, so in this notation we will have as a legal
operation something like `Q(1,2) * M(3,0,-1)` which does not look nice to
me. Plus, `Q` here cannot be a toric lattice in the same sense as `N`
since `N` is unique but it may have different 2-dimensional quotient
lattices with different coercion maps.
* What do you think about representing elements of `Q` by elements of `N`
as well? `Q(N(e))` can either be "the same" as `e`, or we can convert any
input to its "canonical" representative with respect to the fixed lift. I
think the second choice is probably better. Of course, it still will be
possible to get coordinates of elements in a basis of the quotient.
* Should we somehow distinguish these quotient elements in
printing/latexing? In LaTeX it makes sense just to draw a bar over the
whole representative, printing is a bit trickier. Maybe `N/(1,2,3)` or
`N-(1,2,3)`? These do look like arithmetic operations, but since they
should not appear in expressions, it should not be too confusing,
especially the first form with "quotient by a vector".
* Do we care about the choice of the basis for the quotient? I.e. should
there be a possibility to specify it? If not, I think the best way is to
take the same one as is generated by the standard quotient method. (If
yes, we can also choose the basis of dual quotient/sublattice as dual to
the basis of the original sublattice/quotient.)
Let me know what you think,
Andrey
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/9504#comment:3>
Sage <http://www.sagemath.org>
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