Exterior powers of free modules of finite rank

[egourgoulhon]

This ticket implements the exterior power /<sup>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. /<sup>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 /<sup>p(M)
• the element class AlternatingContrTensor for elements of /<sup>p(M)

Note that the pre-existing class for /<sup>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 /<sup>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 /<sup>2(M).

In addition, the ticket implements the interior products

• /<sup>p(M) x /<sup>q(M*) --> /<sup>q-p(M*)
• /<sup>p(M*) x /<sup>q(M) --> /<sup>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.

CC: @tscrim @sagetrac-bpym

Component: linear algebra

Keywords: free module exterior power

Author: Eric Gourgoulhon

Branch/Commit: b797009

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/23207

[egourgoulhon]

Branch: public/manifolds/ext_powers

[egourgoulhon]

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)

[egourgoulhon]

Commit: 8bc9745

[egourgoulhon]

Description changed:

--- 
+++ 
@@ -16,5 +16,5 @@
 
 Besides, some slight updates in all source files in `src/sage/tensor/modules` have been performed:
 
-- a few `range(...)` have been changed to `list(range(...))` to prepare the migration to Python3
+- 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`.

[tscrim]

comment:3

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.

[egourgoulhon]

comment:4

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!

[tscrim]

comment:5

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.

[egourgoulhon]

comment:6

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?

[tscrim]

comment:7

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.

[egourgoulhon]

comment:8

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.