Exterior powers of free modules of finite rank

[Eric Gourgoulhon]

This ticket implements the exterior power /\^p(M) for a free module of finite rank M (p being a positive integer), i.e. the set of alternating contravariant tensors of type (p,0) on M. Previously only the exterior power of the dual of M, i.e. /\^p(M*), was implemented, as the parent of alternating forms of degree p. More specifically, the ticket introduces two new classes:

• the parent class ExtPowerFreeModule for /\^p(M)
• the element class AlternatingContrTensor for elements of /\^p(M)

Note that the pre-existing class for /\^p(M*), which was called ExtPowerFreeModule, has been renamed ExtPowerDualFreeModule, since it regards the exterior power of the dual of M.

The class for the elements of M, FiniteRankFreeModuleElement , inherits from the new class AlternatingContrTensor, reflecting the fact that /\¹(M) = M. In particular, this allows one to consider module elements in operations like the exterior product or the interior product. For instance, for a and b in M, a.wedge(b) returns now the element a/\b of /\²(M).

In addition, the ticket implements the interior products

• /\^p(M) x /\^q(M*) --> /\^q-p(M*)
• /\^p(M*) x /\^q(M) --> /\^q-p(M)

for p<=q via the method interior_product in classes AlternatingContrTensor and FreeModuleAltForm.

Besides, some slight updates in all source files in src/sage/tensor/modules have been performed:

• for the migration to Python3, a few range(...) have been changed to list(range(...)) in places where a list is expected instead of an iterator
• bibliographic references have been reformated to provide links to the master bibliography file src/doc/en/reference/references/index.rst.

[Eric Gourgoulhon]

New commits:

​7451da3	Rename class ExtPowerFreeModule to ExtPowerDualFreeModule
​24437b5	First implementation of exterior powers of free modules (classes ExtPowerFreeModule and AlternatingContrTensor)
​2587df6	Class FiniteRankFreeModuleElement inherits from new class AlternatingContrTensor
​b96b2f3	Add method interior_product in class CompFullyAntiSym
​c58b230	Implement method interior_product in classes AlternatingContrTensor and FreeModuleAltForm
​8bc9745	Complete doctests of FreeModuleAltForm.interior_product + update references (links to master file)

[Travis Scrimshaw]

Quick comment: Rather than antisym=list(range(degree)) (and similar changes), I think it would be better if the Python3 range could be taken as valid input. It makes the code more robust and makes things like list(range(5)) less random in some points of the code.

I will look at the code itself soon.

[Eric Gourgoulhon]

Replying to tscrim:

Quick comment: Rather than antisym=list(range(degree)) (and similar changes), I think it would be better if the Python3 range could be taken as valid input.

This cannot be because the argument antisym is not intended to be an iterator, but a list of integers or tuples encoding the antisymmetries of the tensor; for instance antisym=[(0,1), (2,3)] means that the tensor is antisymmetric with respect to the permutation of its first (0) and second (1) argument/index, as well as with respect to the permutation of its third (2) and fourth (3) argument/index. In this context, antisym=[0,1,2,3] is a short-hand for antisym=[(0,1,2,3)]. For alternating tensors of degree p, one therefore should have antisym=[(0,...,p-1)], short-handed as antisym=[0,...,p-1] and generated by antisym=list(range(p)).

I will look at the code itself soon.

Thanks!

[Travis Scrimshaw]

Replying to egourgoulhon:

Replying to tscrim:

Quick comment: Rather than antisym=list(range(degree)) (and similar changes), I think it would be better if the Python3 range could be taken as valid input.

This cannot be because the argument antisym is not intended to be an iterator, but a list of integers or tuples encoding the antisymmetries of the tensor; for instance antisym=[(0,1), (2,3)] means that the tensor is antisymmetric with respect to the permutation of its first (0) and second (1) argument/index, as well as with respect to the permutation of its third (2) and fourth (3) argument/index. In this context, antisym=[0,1,2,3] is a short-hand for antisym=[(0,1,2,3)]. For alternating tensors of degree p, one therefore should have antisym=[(0,...,p-1)], short-handed as antisym=[0,...,p-1] and generated by antisym=list(range(p)).

range in Python3 has its own type (currently xrange in Python2) that you can explicitly check against:

sage: from six.moves import range as py3range
sage: x = py3range(5)
sage: isinstance(x, py3range)
True

So you can still support the Python3 range but not support iterators.

[Eric Gourgoulhon]

Replying to tscrim:

range in Python3 has its own type (currently xrange in Python2) that you can explicitly check against:

sage: from six.moves import range as py3range
sage: x = py3range(5)
sage: isinstance(x, py3range)
True

So you can still support the Python3 range but not support iterators.

Yes, but I am afraid I don't understand what you have in mind. The attribute _antisym of tensors is a list, which in practice is very short: in most applications, it contains 0 or 1 element, sometimes 2 (e.g. [(0,1), (2,3)] for the fully covariant version of the Riemann tensor). The elements are tuples indicating which indices are involved in the antisymmetry. For the case of alternating tensors, _antisym is a list with a single element, since there is a single antisymmetry, the one corresponding to all possible permutations of the indices. This unique antisymmetry is encoded by the tuple (0,1,...,p-1), p being the degree of the alternating tensor. So we must have _antisym=[(0,1,...,p-1)] in this case, with (0,1,...,p-1) being really a tuple of integers. What would be the advantage of using Python3 range in this context?

[Travis Scrimshaw]

I understand what the input for antisym is suppose to be. I guess the fundamental question is do you want to suppose iterators for the input, whether or not they give the tuples or a integers. I'm just worried about the shock when going to Python3. The option I'm suggesting I guess would be to just automatically run antisym = list(antisym) at the start of the __init__ method. I'm leaning towards the principle of trying to support range whenever possible. However, it is currently not a very strong opinion, so if you don't want to, then it's not something I will hold up this ticket for.

[Eric Gourgoulhon]

Replying to tscrim:

I understand what the input for antisym is suppose to be. I guess the fundamental question is do you want to suppose iterators for the input, whether or not they give the tuples or a integers. I'm just worried about the shock when going to Python3. The option I'm suggesting I guess would be to just automatically run antisym = list(antisym) at the start of the __init__ method.

OK I see.

I'm leaning towards the principle of trying to support range whenever possible. However, it is currently not a very strong opinion, so if you don't want to, then it's not something I will hold up this ticket for.

OK.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​4d24de4

Replace iteritems and itervalues by .items() and .values() in src/sage/tensor/modules

[Eric Gourgoulhon]

The commit in comment:9 gets rid of some six.iteritems and six.itervalues since the lack of iterator is not crucial in the Python2 version; .items() and .values() suffice: no significant loss of performance and this makes the python3_py plugin of the patchbot happy ;-).

[Travis Scrimshaw]

Are you going to support input of range? I think this would be good and means you don't have to wrap so many of those calls with list. I know it is a shorthand, but it still seems very nice to have. Also, I think it would be really good to support

resu.set_comp()[:] = range(1, self._rank+1)

Both _init_derived and _del_derived are unnecessary since they are just super calls.

The changes to tensor_with_indices.py are unnecessary as in Python3:

>>> def foo(a1,a2):
...     print(a1, a2)
... 
>>> foo(*range(2))
0 1

type-(1,0)
type-`(1,0)`

Remove the commented code:

# From sage/modules/module.pyx:
#-----------------------------
### The Element should also implement _rmul_ (or _lmul_)
#
# class MyElement(sage.structure.element.ModuleElement):
#     def _rmul_(self, c):
#         ...

Break the long line here:

     The class :class:`FiniteRankFreeModuleElement` inherits from
-    :class:`~sage.tensor.modules.alternating_contr_tensor.AlternatingContrTensor` because the elements of a free module `M` of
+    :class:`~sage.tensor.modules.alternating_contr_tensor.AlternatingContrTensor`
+    because the elements of a free module `M` of
     finite rank over a commutative ring `R` are identified with tensors of
     type `(1,0)` on `M` via the canonical map

I think this looks better (I believe there are two instances of this):

*p-th exterior power of*
`p`-*th exterior power of*

Are you concerned about dealing with old pickles? If so, have you checked that you can correctly unpickle a previous object? In particular, I think because you renamed the class, the old pickles will not create the correct class, but instead the dual version you constructed here. Although it might just be much more feasible (on our side) to deal with the backwards incompatibility instead.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​41fcf4e

Support of Python3 range in inputs of tensor symmetries

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​59b46cd

Remove unnecessary _init_derived and _del_derived from alternating tensors

[Eric Gourgoulhon]

Replying to tscrim:

Thanks for your comments!

Are you going to support input of range? I think this would be good and means you don't have to wrap so many of those calls with list. I know it is a shorthand, but it still seems very nice to have.

Done! We have now:

sage: from six.moves import range
sage: M = FiniteRankFreeModule(QQ, 3, name='M')
sage: e = M.basis('e')
sage: a = M.tensor((2,1), sym=range(2))
sage: a.symmetries()
symmetry: (0, 1); no antisymmetry
sage: b = M.tensor((0,3), antisym=range(3))
sage: b.symmetries()
no symmetry; antisymmetry: (0, 1, 2)
sage: c = M.tensor((0,3), antisym=range(1,3))
sage: c.symmetries()
no symmetry; antisymmetry: (1, 2)
sage: t = M.tensor((2,0))
sage: t[:] = [[2,1,-3],[0,-4,5],[-1,4,2]]
sage: s = t.symmetrize(*range(2))
sage: s.symmetries()
symmetry: (0, 1); no antisymmetry
sage: s == t.symmetrize(0,1)
True
sage: s = t.antisymmetrize(*range(2))
sage: s.symmetries()
no symmetry; antisymmetry: (0, 1)
sage: s == t.antisymmetrize(0, 1)
True

Also, I think it would be really good to support

resu.set_comp()[:] = range(1, self._rank+1)

Done as well; with the same settings as above, we have:

sage: v = M(range(3))
sage: v.display()
e_1 + 2 e_2
sage: v.set_comp()[:] = range(1, 4)
sage: v.display()
e_0 + 2 e_1 + 3 e_2

Both _init_derived and _del_derived are unnecessary since they are just super calls.

Indeed; I've removed them.

The changes to tensor_with_indices.py are unnecessary as in Python3:

>>> def foo(a1,a2):
...     print(a1, a2)
... 
>>> foo(*range(2))
0 1

Thanks for pointing this out. I've restored the original version of tensor_with_indices.py.

Remove the commented code:

# From sage/modules/module.pyx:
#-----------------------------
### The Element should also implement _rmul_ (or _lmul_)
#
# class MyElement(sage.structure.element.ModuleElement):
#     def _rmul_(self, c):
#         ...

Done.

Break the long line here:

     The class :class:`FiniteRankFreeModuleElement` inherits from
-    :class:`~sage.tensor.modules.alternating_contr_tensor.AlternatingContrTensor` because the elements of a free module `M` of
+    :class:`~sage.tensor.modules.alternating_contr_tensor.AlternatingContrTensor`
+    because the elements of a free module `M` of
     finite rank over a commutative ring `R` are identified with tensors of
     type `(1,0)` on `M` via the canonical map

Done.

I think this looks better (I believe there are two instances of this):

*p-th exterior power of*
`p`-*th exterior power of*

Done.

Are you concerned about dealing with old pickles?

Not at all. I don't think anybody is unpickling old exterior powers of dual free modules. So we can safely ignore this. But thanks for pointing it. One shall indeed have this in mind when changing the name of a class.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​b797009

Typeset of tensor types in the documentation

[Eric Gourgoulhon]

Replying to tscrim:

type-(1,0)
type-`(1,0)`

Done. I've also changed the phrases like of type (0,1) to of type `(0,1)`.

[Travis Scrimshaw]

Great, thank you.

[Eric Gourgoulhon]

Thank you very much for the review!

[Eric Gourgoulhon]

I don't understand why the latest patchbot reports an issue with the .. SEEALSO:: blocks in alternating_contr_tensor.py and free_module_alt_form.py; they seems correctly formed to me and the output html is fine. Moreover, in line 433 of ​https://github.com/sagemath/sage-patchbot/blob/master/sage_patchbot/plugins.py , it is written

1) correct syntax is .. SEEALSO::