Submodules of TensorFreeModule defined by the symmetries of a Components object #30229

mkoeppe · 2020-07-27T01:07:12Z

We generalize FiniteRankFreeModule.tensor_module by giving it optional arguments sym, antisym; if given, a submodule of the tensor module spanned by the tensors with these prescribed symmetries is created.

The new methods symmetric_power and dual_symmetric_power provide two important special cases.

The implementation makes the standard bases of tensor modules (and of their new submodules) explicit objects. The basis method now works for tensor modules, not just the base module, and returns an instance of the new class TensorFreeSubmoduleBasis_sym, which represents the standard basis corresponding to an instance of the Components class (or one of its subclasses).

Follow-ups:

TensorFreeModule.isomorphism_with_fixed_basis #34427 TensorFreeModule.isomorphism_with_fixed_basis
Make ExtPowerFreeModule a quotient of TensorFreeModule #30242 Make ExtPowerFreeModule a quotient of TensorFreeModule
Refactor Components into parent & element #30307 Refactor Components into parent (a ModuleWithBasis) & element

Depends on #30300
Depends on #34424
Depends on #34451
Depends on #34474

CC: @egourgoulhon @tscrim @mjungmath @slel @honglizhaobob

Component: linear algebra

Author: Matthias Koeppe

Branch/Commit: fc66ad1

Reviewer: Eric Gourgoulhon

Issue created by migration from https://trac.sagemath.org/ticket/30229

The text was updated successfully, but these errors were encountered:

egourgoulhon · 2020-07-27T13:26:29Z

comment:2

The current design of ExtPowerFreeModule, ExtPowerDualFreeModule and TensorFreeModule is such that they cannot have their own bases, although they are mathematically free modules and they inherit from FiniteRankFreeModule. This is because all the basis management is performed at the level of the base module. Indeed, any choice of basis in the base module gives birth uniquely to a basis in each of the tensor modules and exterior power modules.
For instance:

sage: M = FiniteRankFreeModule(ZZ, 2, name='M')
sage: e = M.basis('e')
sage: t = M.tensor((1, 1), name='t'); t
Type-(1,1) tensor t on the Rank-2 free module M over the Integer Ring

The components are set and displayed with respect to the basis e = (e_0, e_1) of the base module M and its dual basis (e^0, e^1):

sage: t[e, :] = [[1, 0], [-2, 0]]
sage: t.display(e)  # equivalent to t.display()
t = e_0*e^0 - 2 e_1*e^0
sage: t.components(e)
2-indices components w.r.t. Basis (e_0,e_1) on the Rank-2 free module M 
 over the Integer Ring

The same feature holds for ExtPowerDualFreeModule elements:

sage: a = M.alternating_form(2, name='a')
sage: a.parent()
2nd exterior power of the dual of the Rank-2 free module M over the Integer Ring
sage: a[e, 0, 1] = 3
sage: a.display(e)  # equivalent to a.display()
a = 3 e^0/\e^1
sage: a.components(e)
Fully antisymmetric 2-indices components w.r.t. Basis (e_0,e_1) on the Rank-2 
 free module M over the Integer Ring

If we introduce proper bases for the derived modules ExtPowerFreeModule, ExtPowerDualFreeModule and TensorFreeModule, then t.display() would become ambiguous: shall we want a display w.r.t. to the default basis of M or to the default basis of t.parent()? Moreover, the storage of components is currently performed with respect to bases of the base module. Introducing bases in the tensor modules shall of course not lead to duplicate storage.

If we keep this design (which I would favor at this stage), then, for the sake of clarity,

sage: A = M.exterior_power(2)
sage: A.basis('w')

shall return a NotImplementedError and not a TypeError, with a message like only bases on the base module are implemented.

mkoeppe · 2020-07-27T15:59:52Z

comment:4

Thanks a lot for the explanation.

Yes, a NotImplementedError with a clear message would be an improvement.

It could also be considered whether these three classes should really be subclasses of FiniteRankFreeModule, or perhaps rather a new common baseclass.

egourgoulhon · 2020-07-27T17:39:30Z

comment:6

Replying to @mkoeppe:

It could also be considered whether these three classes should really be subclasses of FiniteRankFreeModule, or perhaps rather a new common baseclass.

Yes, indeed!

mkoeppe · 2020-07-27T17:42:39Z

comment:7

Replying to @egourgoulhon:

all the basis management is performed at the level of the base module. Indeed, any choice of basis in the base module gives birth uniquely to a basis in each of the tensor modules and exterior power modules.

Given this, I am also wondering whether perhaps there should be a subclass of Basis_abstract that would make these bases explicit.

mkoeppe · 2020-07-31T03:00:11Z

Branch: u/mkoeppe/extpowerfreemodule__extpowerdualfreemodule__tensorfreemodule_cannot_create_a_basis

mkoeppe · 2020-07-31T03:02:57Z

comment:9

Preliminary branch

New commits:

`862aca8`	`WIP: Implement TensorFreeModule.basis with new class FreeModuleCompTensorBasis`

mkoeppe · 2020-07-31T03:02:57Z

Commit: 862aca8

sagetrac-git · 2020-08-01T18:11:48Z

Changed commit from 862aca8 to 3bd0a8a

sagetrac-git · 2020-08-01T18:11:48Z

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

`3bd0a8a`	`TensorFreeSubmodule_comp, TensorFreeSubmoduleBasis_comp: New`

sagetrac-git · 2020-08-01T18:38:48Z

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

`c4d8d07`	`Move basis method to TensorFreeModule`

sagetrac-git · 2020-08-01T18:38:48Z

Changed commit from 3bd0a8a to c4d8d07

mkoeppe · 2020-08-01T18:39:24Z

comment:13

Here's an early draft, comments welcome

egourgoulhon · 2020-08-01T20:08:22Z

comment:14

Replying to @mkoeppe:

Here's an early draft, comments welcome

Thanks. It seems that the file tensor_free_submodule_basis.py is missing in the ticket branch.

sagetrac-git · 2020-08-01T20:12:38Z

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

`270ebb3`	`Add missing file`

sagetrac-git · 2020-08-01T20:12:38Z

Changed commit from c4d8d07 to 270ebb3

mkoeppe · 2020-08-01T20:13:05Z

comment:16

Sorry, here you go.

sagetrac-git · 2020-08-03T04:21:31Z

Changed commit from 270ebb3 to 22bb5ac

sagetrac-git · 2020-08-03T04:21:31Z

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

`a734bf8`	`TensorFreeSubmodule_comp, TensorFreeSubmoduleBasis_comp: New`
`fdd6646`	`Move basis method to TensorFreeModule`
`a61f8e7`	`Add missing file`
`c01a5e9`	`FiniteRankFreeModule: Add methods is_submodule, ambient_module`
`1aedfa4`	`TensorFreeModule: Remove duplicate method 'basis'`
`22bb5ac`	`TensorFreeSubmodule_comp.is_submodule: New`

sagetrac-git · 2020-08-03T04:28:03Z

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

`6ed9e40`	`TensorFreeSubmodule_comp.ambient_module: New`
`80af815`	`Fix doctests`

sagetrac-git · 2022-09-02T17:35:22Z

Changed commit from 64e3c1d to 7474abb

sagetrac-git · 2022-09-02T17:35:22Z

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

`7474abb`	`FiniteRankFreeModule_abstract.tensor_product: Handle submodules with symmetries`

sagetrac-git · 2022-09-02T17:37:35Z

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

`ed3c8d9`	`src/sage/tensor/modules/finite_rank_free_module.py (FiniteRankFreeModule_abstract.is_submodule): Fix typo in doctest`

sagetrac-git · 2022-09-02T17:37:35Z

Changed commit from 7474abb to ed3c8d9

sagetrac-git · 2022-09-02T17:48:28Z

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

`fd89892`	`src/sage/tensor/modules/tensor_free_submodule[_basis].py, finite_rank_free_module.py: Update AUTHORS`
`644933a`	`src/sage/tensor/modules/finite_rank_free_module.py: Update copyright according to git blame -w --date=format:%Y FILE \| sort -k2`

sagetrac-git · 2022-09-02T17:48:28Z

Changed commit from ed3c8d9 to 644933a

sagetrac-git · 2022-09-02T18:06:24Z

Changed commit from 644933a to 45bf6f3

sagetrac-git · 2022-09-02T18:06:24Z

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

`45bf6f3`	`FiniteRankFreeModule_abstract.tensor_product: Add comment`

egourgoulhon · 2022-09-04T17:05:33Z

comment:120

Looks good! Thanks.

egourgoulhon · 2022-09-04T17:05:33Z

Reviewer: Eric Gourgoulhon

mkoeppe · 2022-09-04T18:08:55Z

comment:121

Thank you!

sagetrac-git · 2022-09-04T18:45:53Z

Changed commit from 45bf6f3 to fc66ad1

sagetrac-git · 2022-09-04T18:45:53Z

Branch pushed to git repo; I updated commit sha1 and set ticket back to needs_review. New commits:

`803f7e4`	`Make FiniteRankFreeModule.tensor_module(0, 1) return the dual (#34474)`
`0648daa`	`Merge #34474`
`fc66ad1`	`FiniteRankFreeModule.dual_symmetric_power: Update for changes in #34474`

mkoeppe · 2022-09-04T18:46:41Z

Changed dependencies from #30300, #34424, #34451 to #30300, #34424, #34451, #34474

mkoeppe · 2022-09-04T18:46:41Z

comment:123

I've merged #34474 to resolve a merge conflict

New commits:

`803f7e4`	`Make FiniteRankFreeModule.tensor_module(0, 1) return the dual (#34474)`
`0648daa`	`Merge #34474`
`fc66ad1`	`FiniteRankFreeModule.dual_symmetric_power: Update for changes in #34474`

egourgoulhon · 2022-09-04T19:19:01Z

comment:124

Replying to Matthias Köppe:

I've merged #34474 to resolve a merge conflict

OK.

At the moment, we have:

sage: M = FiniteRankFreeModule(ZZ, 2, name='M')
sage: e = M.basis('e')
sage: S = M.tensor_module(2, 0, sym=(0,1))
sage: s = M.tensor((2, 0), sym=(0,1))
sage: s[0,1] = 2
sage: s.display()
2 e_0⊗e_1 + 2 e_1⊗e_0
sage: s.parent() is S
False
sage: s.parent()
Free module of type-(2,0) tensors on the Rank-2 free module M over the Integer Ring

but I guess s.parent() being S should be for a follow-up ticket...

mkoeppe · 2022-09-04T19:21:01Z

comment:125

Yes, #34482 goes roughly into this direction.

egourgoulhon · 2022-09-04T19:28:41Z

comment:126

Replying to Matthias Köppe:

Yes, #34482 goes roughly into this direction.

OK very good.

vbraun · 2022-09-22T22:35:02Z

Changed branch from u/mkoeppe/extpowerfreemodule__extpowerdualfreemodule__tensorfreemodule_cannot_create_a_basis to fc66ad1

mkoeppe added this to the sage-9.2 milestone Jul 27, 2020

mkoeppe added c: linear algebra labels Jul 27, 2020

mkoeppe changed the title ~~ExtPowerFreeModule and TensorFreeModule cannot create a basis~~ ExtPowerFreeModule, ExtPowerDualFreeModule, TensorFreeModule cannot create a basis Jul 27, 2020

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mkoeppe changed the title ~~ExtPowerFreeModule, ExtPowerDualFreeModule, TensorFreeModule cannot create a basis~~ Submodules of TensorFreeModule defined by the symmetries of a Components object Aug 1, 2020

egourgoulhon added s: positive review and removed s: needs review labels Sep 4, 2022

sagetrac-git mannequin added s: needs review and removed s: positive review labels Sep 4, 2022

egourgoulhon added s: positive review and removed s: needs review labels Sep 4, 2022

mkoeppe modified the milestones: sage-9.7, sage-9.8 Sep 19, 2022

vbraun removed the s: positive review label Sep 22, 2022

vbraun closed this as completed in aa22b33 Sep 22, 2022

egourgoulhon mentioned this issue Jan 30, 2023

SageManifolds metaticket 2 #30525

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Submodules of TensorFreeModule defined by the symmetries of a Components object #30229

Submodules of TensorFreeModule defined by the symmetries of a Components object #30229

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