Representation theory of finite dimensional associative algebras #16659

avirmaux · 2014-07-16T01:55:10Z

This ticket implements representation theory methods for a finite
dimensional associative algebra A:

Maximal decomposition of the identity into primitive idempotents if
A is semi-simple and commutative;
Maximal decomposition of the identity into a sum of central
orthogonal idempotents if A is semi simple;
Lifting thereof from the semisimple quotient of A;
Peirce decomposition of A;
Information on the simple A-mod and indecomposable projective A-mod,
and in particular the Cartan invariant matrix of A.
projective indecomposable modules (up to multiplicity)
principal ideals

Also optimizes Algebras.FiniteDimensional.ParentMethods.radical_basis, and remove now useless cached_product option since products are always cached.

See #18311 for a follow up.

Depends on #11111

CC: @nthiery @nathanncohen @simon-king-jena

Component: algebra

Keywords: representation theory, days64, sd67

Author: Aladin Virmaux

Branch: ff577f3

Reviewer: Franco Saliola, Nicolas M. Thiéry

Issue created by migration from https://trac.sagemath.org/ticket/16659

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

avirmaux · 2014-07-25T01:25:42Z

Branch: u/virmaux/t/16659

sagetrac-git · 2014-07-25T01:31:31Z

Commit: f94621e

sagetrac-git · 2014-07-25T01:31:31Z

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

`f94621e`	`indentation`

sagetrac-git · 2014-07-25T02:36:13Z

Changed commit from f94621e to a203460

sagetrac-git · 2014-07-25T02:36:13Z

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

`f0bb894`	`_mul_ at the right place`
`a3b0987`	`Merge branch 't/11111/public/11111-more_support_for_finite_dimensional_modules_and_algebras_with_basis' into t/16659`
`a203460`	`tests`

sagetrac-git · 2014-07-25T18:34:24Z

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

`75e4f5a`	`_lift_idempotent for elements of sub-algebras`
`7caba0d`	`orthogonal idempotents for semisimple algebras`
`8aa7db9`	`tests`
`b2bb566`	`beginning of implemantation for general algebras`

sagetrac-git · 2014-07-25T18:34:24Z

Changed commit from a203460 to b2bb566

sagetrac-git · 2014-08-29T09:06:07Z

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

`a156a1e`	`radical for SemisimpleAlgebras`
`ebb8cb5`	`example`
`f01ecd1`	`orthogonal idempotents for all finitedimensional algebras with basis`

sagetrac-git · 2014-08-29T09:06:07Z

Changed commit from b2bb566 to f01ecd1

sagetrac-git · 2014-08-29T09:08:39Z

Changed commit from f01ecd1 to 57fbdc0

sagetrac-git · 2014-08-29T09:08:39Z

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

`57fbdc0`	`doctest`

sagetrac-git · 2014-08-29T10:13:30Z

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

`aa1a19c`	`example lift_idempotent`
`d107ce1`	`Cartan invariant matrix for general finite dimension algebras with basis`

sagetrac-git · 2014-08-29T10:13:30Z

Changed commit from 57fbdc0 to d107ce1

sagetrac-git · 2014-08-29T10:36:27Z

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

`9486c19`	`projective modules`

sagetrac-git · 2014-08-29T10:36:27Z

Changed commit from d107ce1 to 9486c19

sagetrac-git · 2014-08-29T10:53:50Z

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

`4efa877`	`projective decomposition`

sagetrac-git · 2014-08-29T10:53:50Z

Changed commit from 9486c19 to 4efa877

pjbruin · 2014-08-29T11:07:15Z

comment:11

This may be a superfluous comment, but have you taken a look at #12141 (merged 6 months ago) and have you checked that you are not duplicating any of the functionality there? (I'm asking because you seem to work mostly inside sage.categories and not to interact with sage.algebras.)

sagetrac-git · 2014-09-01T11:45:08Z

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

`fe0dfca`	`merge`
`44e8606`	`merge conflict`

sagetrac-git · 2014-09-01T11:45:08Z

Changed commit from 4efa877 to 44e8606

avirmaux · 2014-09-01T12:01:01Z

comment:13

pbruin: thanks for the hint, I will definitely check :)

sagetrac-git · 2014-09-01T14:16:08Z

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

sagetrac-git · 2014-09-01T14:16:08Z

Changed commit from 44e8606 to 4efa877

sagetrac-git · 2014-09-01T16:15:21Z

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

`016cd16`	`#16340: Revert ReST typo fix in ell_curve_isogeny.py to avoid conflict with other ticket handling it`
`d147678`	`Merge branch 'develop=6.4.beta0' into categories/morphism-methods-10668`
`8054997`	`Merge branch 'categories/morphism-methods-10668' into categories/module-morphisms-8678`
`09964af`	`Merge branch 'categories/module-morphisms-8678' into t/11111/more_support_for_finite_dimensional_modules_and_algebras_with_basis`
`195d14e`	`Merge branch 'public/11111-more_support_for_finite_dimensional_modules_and_algebras_with_basis' of trac.sagemath.org:sage into t/11111/more_support_for_finite_dimensional_modules_and_algebras_with_basis`
`1332742`	`Merge branch 't/11111' into t/16659`
`9c4d13d`	`11111: revert subobjects of algebras`
`d1fc0a9`	`11111: the center of a semisimple algebra is semisimple.`
`c6eebd2`	`merge 11111`
`7cbeb9a`	`16659: renamed, moved some methods to better places`

nthiery · 2015-04-27T14:32:47Z

comment:91

Replying to @saliola:

I would love to take a look at the code and play around with it a bit more. I can do it by Friday, if you don't mind waiting until then.

Sure thing! Thanks in advance.

nthiery · 2015-04-27T17:02:08Z

comment:92

Gasp, I just noticed that I put a wrong ticket number (#17696 instead of #16659) in some of the commit messages.

saliola · 2015-05-01T16:57:42Z

comment:93

Check it out! I'm actually accomplishing this by my self-imposed deadline. It's rare when this happens. :-)

First of all, this looks great! I only have a few comments / questions.
orthogonal_idempotents_central_mod_rad: Isn't this the same thing as a complete set of primitive orthogonal idempotent? If so then I suggest the name complete_set_of_primitive_orthogonal_idempotents. I think this is the standard name in the literature. Maybe I'm missing something? (If we stick with the original name, then I suggest radical instead of rad; the name is long so a few more characters won't be a problem)
idempotent_lift: there are other possible lifts one could use here. Is it worth implementing some of these others and giving the user a choice? (doesn't have to be done on this ticket; just something I'd like to hear about)
cartan_invariants_matrix: in the documentation it is written that S_i and their adjoints S_i^* are representatives of the right simple modules; did you mean left for the latter?
Tests: All tests passed!

Positive review pending the change of right to left for S_i^*.

saliola · 2015-05-01T17:03:52Z

comment:94

Replying to @saliola:

idempotent_lift: there are other possible lifts one could use here. Is it worth implementing some of these others and giving the user a choice? (doesn't have to be done on this ticket; just something I'd like to hear about)

For instance, it seems that one can iterate the formula 3 * a^2 - 2 * a^3 to lift the idempotents. See Lemma 1.7.3 (Idempotent Refinement) of Benson's 'Representations and Cohomology I (search for refinement).

nthiery · 2015-05-02T06:46:48Z

comment:95

Replying to @saliola:

Check it out! I'm actually accomplishing this by my self-imposed deadline. It's rare when this happens. :-)

Quite impressed, I am!

:-)

First of all, this looks great! I only have a few comments / questions.

Thanks, and thanks for the review!

orthogonal_idempotents_central_mod_rad: Isn't this the same thing as a complete set of primitive orthogonal idempotent?

Nope; in the case of a full nxn matrix algebra (which is semisimple,
so mod rad or not are the same), we get a single idempotent: the
identity; to get primitive idempotents, we would need to refine this
trivial decomposition into n orthogonal idempotents.

In general, for each simple module of dim k, we get a single
idempotent e such that eAe is the k x k block matrix corresponding to
the simple module.

We haven't found a good name for those idempotents in the literature;
suggestions welcome!

(If we stick with the original name, then I suggest radical instead of rad; the name is long so a few more characters won't be a problem)

Will do!

idempotent_lift: there are other possible lifts one could use here. Is it worth implementing some of these others and giving the user a choice? (doesn't have to be done on this ticket; just something I'd like to hear about)

If we have a use case for it, that is if the various lifting have each
some specific merit, then why not. For a latter ticket indeed.

cartan_invariants_matrix: in the documentation it is written that S_i and their adjoints S_i^* are representatives of the right simple modules; did you mean left for the latter?

Oops. Will fix now!

Cheers,
Nicolas

sagetrac-git · 2015-05-02T06:52:24Z

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

`5f18523`	`Merge branch 'develop' into representation_theory/finite_dimensional_algebras-16659`
`661c008`	`16659: orthogonal_idempotents_central_mod_rad -> ...mod_radical + small doc fix`

sagetrac-git · 2015-05-02T06:52:24Z

Changed commit from faf83a1 to 661c008

nthiery · 2015-05-02T06:53:31Z

comment:97

Implemented final fix requested by Franco; positive review on his behalf!

New commits:

`5f18523`	`Merge branch 'develop' into representation_theory/finite_dimensional_algebras-16659`
`661c008`	`16659: orthogonal_idempotents_central_mod_rad -> ...mod_radical + small doc fix`

vbraun · 2015-05-02T17:21:07Z

comment:98

Overfull \hbox (49.85445pt too wide) in paragraph at lines 44441--44442
\T1/ptm/m/n/10 Bases: \T1/pcr/m/n/10 sage.categories.category_with_axiom.Catego
ryWithAxiom_over_base_ring 

! LaTeX Error: Too deeply nested.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.44470 \begin{itemize}
                       
? >
! Emergency stop.
 ...                                              
                                                  
l.44470 \begin{itemize}
                       
!  ==> Fatal error occurred, no output PDF file produced!
Transcript written on categories.log.

tscrim · 2015-05-03T06:54:59Z

comment:99

Easiest way to fix this is probably to just pull out SemisimpleAlgebras.FiniteDimensional.WithBasis into a separate class (it seems big enough to warrant this anyways IMO).

nthiery · 2015-05-03T08:20:10Z

comment:100

Good morning Travis,

Replying to @tscrim:

Easiest way to fix this is probably to just pull out SemisimpleAlgebras.FiniteDimensional.WithBasis into a separate class (it seems big enough to warrant this anyways IMO).

Perfect, that's exactly what I was working on :-)

Great minds ...

sagetrac-git · 2015-05-03T08:24:46Z

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

`ea552de`	`16659: split Algebras.Semisimple.FiniteDimensional.WithBasis in its own file: first step: copying the file`
`ff577f3`	`16659: split Algebras.Semisimple.FiniteDimensional.WithBasis in its own file: actual work`

sagetrac-git · 2015-05-03T08:24:46Z

Changed commit from 661c008 to ff577f3

nthiery · 2015-05-03T08:28:31Z

comment:102

Done!

All long tests pass in sage/categories, and the pdf doc
compiles. I haven't made any change besides moving the code, and
adding the usual file/class docstrings, lazy imports, and addition to
the ref manual. If someone can briefly double check those, I believe
this can be set back to positive review.

nthiery · 2015-05-03T08:41:04Z

comment:103

Replying to @vbraun:

! LaTeX Error: Too deeply nested.

Thanks Volker for the report, and sorry for not having compiled the
pdf doc!

To reduce the risk of this happening again, I have posted a feature
request for the patchbot to build and report on the pdf doc:

sagemath/sage-patchbot#64

Cheers,
Nicolas

vbraun · 2015-05-03T14:53:30Z

Changed branch from public/representation_theory/finite_dimensional_algebras-16659 to ff577f3

nthiery · 2015-05-03T20:28:52Z

Changed commit from ff577f3 to none

nthiery · 2015-05-03T20:28:52Z

comment:106

Thanks Franco and Volker for this final step!

Yeah, this is done! One more big feature of MuPAD-Combinat that is in Sage :-)

avirmaux mannequin added this to the sage-6.3 milestone Jul 16, 2014

avirmaux mannequin added c: algebra labels Jul 16, 2014

sagetrac-vbraun-spam mannequin modified the milestones: sage-6.3, sage-6.4 Aug 10, 2014

nthiery added s: positive review and removed s: needs review labels May 2, 2015

vbraun added s: needs work and removed s: positive review labels May 2, 2015

nthiery added s: needs review and removed s: needs work labels May 3, 2015

saliola added s: positive review and removed s: needs review labels May 3, 2015

vbraun removed the s: positive review label May 3, 2015

vbraun closed this as completed in 61311c4 May 3, 2015

This was referenced Dec 29, 2022

Representation theory of finite semigroups #12914

Open

Finite dimensional modules with basis: improved conversions to vectors and matrices #18310

Open

Representation theory of finite dimensional associative algebras #16659

Representation theory of finite dimensional associative algebras #16659

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avirmaux mannequin commented Jul 16, 2014

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