Implement Specht modules in the tabloid basis

[tscrim]

This provides an implementation of the Specht modules over arbitrary fields in the standard tableaux basis. To do so, we also implement the tabloid module, where the tabloids are implemented as OrderedSetsPartitions. Using this, we can construct the maximal submodule and simple submodule from a Specht module.

We also implement a method to compute the Brauer character of the representations.

📝 Checklist

• The title is concise, informative, and self-explanatory.
• The description explains in detail what this PR is about.
• I have linked a relevant issue or discussion.
• I have created tests covering the changes.
• I have updated the documentation accordingly.

⌛ Dependencies

• Compute dimensions of simple symmetric group modules over positive characteristic #36724: Trivial conflict with working on the same files.
• ModulesWithBasis.ParentMethods.cardinality() for 0 dimensional modules #36985: So the 0 dimensional tests will pass.

[tscrim]

I have addressed all of the comments and added a few more doctests for the codecov (those I felt were reasonable).

[tscrim]

@darijgr ping?

[mantepse]

@tscrim, is there any reason you know of that this code is not yet in sage?

[darijgr]

@mantepse Lazy reviewers...

[mantepse]

@darijgr, do you want to look more thoroughly? For me it certainly looks "good enough" - I would be extremely surprised to see any fundamental flaws that cannot be fixed in follow up tickets. What do you think?

[darijgr]

I don't have time any more in this quarter :(

Martin, can you review this?

[mantepse]

Thank you!

(I'll leave it up to you whether you want to allow compositions now or not)

[mantepse]

Thank you!

[mantepse]

Copying this from above, because apparently, once resolved, text is not shown anymore.

I agree. In fact, I now think that Tabloids should really be OrderedSetPartitions_scomp(n, alpha), without any extra methods. Maybe the name of the method symmetric_group_action is not ideal, do we need it at all?

For example, __iter__ of OrderedSetPartitions_scomp should be the same as __iter__, so we don't need to optimize two iterators.

I'll give it positive review anyway and leave this to you.

[tscrim]

Copying this from above, because apparently, once resolved, text is not shown anymore.

I agree. In fact, I now think that Tabloids should really be OrderedSetPartitions_scomp(n, alpha), without any extra methods. Maybe the name of the method symmetric_group_action is not ideal, do we need it at all?

I have two reasons for this:

1. I use the symmetric_group_action method to build the action. Although this could be moved into the tabloid module itself.
2. I want to actually have latex/ascii_art output as tabloids. This can also be moved into the tabloid module code, but it is a less natural place.

For example, __iter__ of OrderedSetPartitions_scomp should be the same as __iter__, so we don't need to optimize two iterators.

Indeed.

I'll give it positive review anyway and leave this to you.

Thank you, but I seriously considering dumping the Tabloids class for OSPs. Its duplicating too much for no clear benefits or extensions that OSPs wouldn't be able to do themselves (with perhaps a special subclass for those whose base set is $[n]$). It seems like you are also in favor of this as well, right?

[mantepse]

Yes, I am always in favour of less duplication.

[mantepse]

PS: I think eventually we should have a clear framework for combinatorial objects 'naturally defined on the set [n]' and on arbitrary sets, that is used consistently for SetPartitions, Combinations, Tableaux, etc.

(and also walk through possibly misleading classes, eg., Tuples, and eventually deprecate them)

[tscrim]

I have removed the Tabloid(s) classes in my latest push since they are (currently) redundant and moved the corresponding functionality into the TabloidModule class. I also fixed a bug with partitions not being recognized as compositions (necessary for the code now). I reshuffled the class hierarchy a bit in order to bring in more functionality from Representation_abstract. To get everything to work, I needed to add a gens() method to StandardPermutations_n.

[fchapoton]

the doctests, they are a failin' 🎶

[tscrim]

Seems like how the error is generated changed with the changes to the submodule code. I will need to check this before updating the error message; although I feel like the error message should not be changed...

[tscrim]

Come gather 'round people, wherever you roam, and finish the PR review with doctest passin'.

So it was a deeper issue (and more strangely/worrisome, something I couldn't reproduce locally!). The is_submodule check should not have raised an error if the ambient spaces were different but instead simply returned False because of the generic coercion check for modules is relying on that. This is now fixed in the last commit and the doctests in specht_module.py now pass for me.

[github-actions]

Documentation preview for this PR (built with commit 95a8ed3; changes) is ready! 🎉

[fchapoton]

ok, let's go

[tscrim]

Merci!