The Burnside ring of a group

[Newtech66]

Resolves #35475.

We implement the Burnside ring for a group $G$. We provide a few ways to construct elements, either directly as a formal sum of orbit types $[G/H]$ (where $H$ is a subgroup of $G$), or by providing a group action $\alpha: G \times X \rightarrow X$ on some set $X$. Finally, we implement addition and multiplication of ring elements.

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@mantepse

[github-actions]

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[mantepse]

@trevorkarn, I'm adding you here because of your request at #35475 (comment)

The current branch is a prototype, not yet ready for review.

[trevorkarn]

@trevorkarn, I'm adding you here because of your request at #35475 (comment)

The current branch is a prototype, not yet ready for review.

Thanks so much @mantepse! Just let me know when it is ready for review.

[mantepse]

I just realised that you can remove BurnsideRing._element_constructor_ completely, and instead implement ConjugacyClassesOfSubgroups.__contains__ (which we should provide anyway):

    def __contains__(self, H):
        return H.is_subgroup(self._G)

How is that working under the hood?

Not sure what exactly you mean here.

1. __contains__ is python magic https://docs.python.org/3/reference/datamodel.html#object.__contains__
2. CombinatorialFreeModule._element_constructor_ uses __contains__ here:
   

   sage/src/sage/combinat/free_module.py

   Lines 763 to 764 in ffbbea9

   	elif x in self._indices:
   	return self.monomial(self._indices(x))

[Newtech66]

1. __contains__ is python magic https://docs.python.org/3/reference/datamodel.html#object.__contains__
2. CombinatorialFreeModule._element_constructor_ uses __contains__ here:
   

   sage/src/sage/combinat/free_module.py

   Lines 763 to 764 in ffbbea9

   	elif x in self._indices:
   	return self.monomial(self._indices(x))

Ah, I see, that makes sense.

[Newtech66]

I propose we implement the conjugacy class renaming like B[Z3].rename_index("Z3"). We can make it fail if we try to call it on something that is not a basis element. Any alternative suggestions? Can the function be named better?

[mantepse]

I propose we implement the conjugacy class renaming like B[Z3].rename_index("Z3"). We can make it fail if we try to call it on something that is not a basis element. Any alternative suggestions? Can the function be named better?

Maybe, I don't know. I think we can delay thinking about this until we start to play with species.

[mantepse]

Suggested change

	return H.is_subgroup(self._G)
	if parent(H) == self:
	return True
	return (isinstance(H, PermutationGroup_generic)
	and H.is_subgroup(self._G))

[Newtech66]

from sage.structure.element import parent as get_parent_of was required, then if get_parent_of(H) == self.

[mantepse]

Oh, sorry, yes. Except the renaming, this is unnecessary.

[dimpase]

Does this use tables of marks? (With which it's much easier). GAP does have functionality to deal with tables of marks,
and has a large library of them, packaged as tomlib package. (tomlib is a part of Sage)

[mantepse]

Does this use tables of marks? (With which it's much easier). GAP does have functionality to deal with tables of marks, and has a large library of them, packaged as tomlib package. (tomlib is a part of Sage)

Not yet, but eventually we will use it whenever available. However, as you certainly know, the table of marks works is only available for relatively small groups, e.g. the symmetric groups up to $\mathfrak S_{13}$. I hope that some operations will be possible even for larger groups, although, of course, they will be very slow.

[Newtech66]

Why are we adding SymmetricGroupAction in the name? The molecular decomposition of $F[n]$ is defined wrt $\mathfrak{S}_n$, so why the additional qualifier?

[mantepse]

Why are we adding SymmetricGroupAction in the name? The molecular decomposition of $F[n]$ is defined wrt $\mathfrak S_n$, so why the additional qualifier?

You are right, PolynomialMolecularDecomposition would be better. I admit, I don't care much about the name for the moment, and I didn't think much about it.

For me, the only question is whether we need this class at all, and I think that we do.

[Newtech66]

If we use the GradedAlgebrasWithBasis category then completion/formal_series_ring should be provided automatically.

[mantepse]

If we use the GradedAlgebrasWithBasis category then completion/formal_series_ring should be provided automatically.

Yes, that's correct, and that's the whole point of having a class PolynomialMolecularDecomposition (or whatever name we use). Is it clear to you how to implement it? It will inherit again from CombinatorialFreeModule, the only part which might involve work is to get the basis_keys right. Possibly you have to define a class for these, i.e., ConjugacyClassesOfSymmetricGroupSubgroups - which is in InfiniteEnumeratedSets.