remove adams_operation

[zabrocki]

adams_operator was added at the level of bialgebras and is defined as \mu^{n-1} \circ \Delta^{n-1} however in symmetric functions the method adams_operation is an alias to frobenius. Since this is confusing. We propose removing adams_operation as an alias for frobenius. adams_operation was added in #14775, adams_operator will be added in #18678.

The documentation explains: The n-th Frobenius operator is also the n-th Adams operator of the \Lambda-ring of symmetric functions over the integers. This does not seem to agree with what we found in a paper by Aguiar and Lauve "The characteristic polynomial of the Adams operators on graded connected Hopf algebras" which says the Adams operators are the "Hopf powers or Sweedler powers" and it gives the definition that was defined at the level of bialgebras.

[darij]

Good point about the conflicting terminology. I don't mind removing these aliases (but I fear I have no time whatsoever to actually do any of the job myself).

[darij]

New commits:

​c75f51d

remove the line 'adams_operation = frobenius' (and comments about the adams operation in frobenius method)

[darij]

Please keep the comment about the Adams operator of the lambda-ring. That at least is correct.

Also there is a similar problem with QSym.

[zabrocki]

Thanks Darij. Jean-Baptiste is here visiting and we wanted to check with you that it is ok to remove the aliases. Can you explain how the terminology is correct? What are we taking as the definition of "Adams operator of the lambda-ring"?

[darij]

Every lambda-ring canonically has Adams operations defined on it ( ​https://www.encyclopediaofmath.org/index.php/Lambda-ring ). The symmetric functions have a canonical lambda-ring structure. (There is yet another one on their homogeneous components, but I'm talking about the one on the whole ring.) The Adams operations of this lambda-ring structure are the Frobenius operators. This is in Hazewinkel's various texts.

I think there are good reasons for Aguiar and Lauve calling their operators "Adams operators", but I don't think these reasons come from lambda-ring theory.

[zabrocki]

I see now. Both operations are Adams operators. frobenius is the Adam's operator for the bialgebra p_r \circ p_n = p_{rn} and \Delta^{\circ}(p_n) = p_n \otimes p_n. Thanks. We will try to make that clear.

[git]

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

​233a05c

remove methods adams_operator to avoid conflict with the natural Adams operator of the bialgebra

[darij]

LGTM. Do the doctests agree?

I positively hope noone has used the aliases yet, since I have no idea how deprecation works.

[zabrocki]

Good point. It will take a while to deprecate the aliases, but long term its better not to have both adams_operation and adams_operator

[git]

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

​3264c67

add deprecation of adams_operation methods in sf and qsym

[darij]

+        The Frobenius operator is the Adam's operator for the bialgebra
+        `p_r \circ p_n = p_{rn}` and `\Delta^{\circ}(p_n) = p_n \otimes p_n`.

That's wrong. I think the n-th Adams operator of this bialgebra would send p_k to p_{k^n}. I also don't think that this is a good way to understand the Frobenius.

[git]

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

​5f785a2

update comment following the darij's recommandations

[darij]

LGTM. If the tests run, this ticket is good. Thanks for the disambiguation!

[tscrim]

Tests pass for me, so given Darij's comments, positive review.