Opened 7 years ago

Closed 7 years ago

## #19255 closed defect (fixed)

Reported by: Owned by: zabrocki minor sage-6.9 combinatorics alauve, darij, aschilling, tscrim, nthiery, hivert, sage-combinat, mhansen, elixyre Mike Zabrocki, Jean-Baptiste Priez Darij Grinberg, Travis Scrimshaw N/A 5f785a2 5f785a2df1f0f3fc033fe7e5d933914fcf7d43e4

### Description

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.

### comment:2 Changed 7 years ago by 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).

### comment:3 Changed 7 years ago by darij

New commits:

 ​c75f51d remove the line 'adams_operation = frobenius' (and comments about the adams operation in frobenius method)

### comment:4 Changed 7 years ago by darij

Also there is a similar problem with QSym.

### comment:5 Changed 7 years ago by 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"?

### comment:6 Changed 7 years ago by 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.

### comment:7 Changed 7 years ago by 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.

### comment:8 Changed 7 years ago by git

• Commit changed from c75f51d6a4a3e69a6d3ad95ecd51cb7f09caebd1 to 233a05c9655fa90b297109944b9c96345aa8fd55

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

### comment:9 Changed 7 years ago by darij

LGTM. Do the doctests agree?

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

### comment:10 Changed 7 years ago by 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

### comment:11 Changed 7 years ago by git

• Commit changed from 233a05c9655fa90b297109944b9c96345aa8fd55 to 3264c679c434df7b87bbe01d6d887f0175c434f9

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

 ​3264c67 add deprecation of adams_operation methods in sf and qsym

### comment:12 Changed 7 years ago by elixyre

• Authors changed from elixyre to Mike Zabrocki, Jean-Baptiste Priez

### comment:13 Changed 7 years ago by elixyre

• Status changed from new to needs_review

### comment:14 Changed 7 years ago by 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.

### comment:15 Changed 7 years ago by git

• Commit changed from 3264c679c434df7b87bbe01d6d887f0175c434f9 to 5f785a2df1f0f3fc033fe7e5d933914fcf7d43e4

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

 ​5f785a2 update comment following the darij's recommandations

### comment:16 Changed 7 years ago by darij

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

### comment:17 Changed 7 years ago by tscrim

• Reviewers set to Darij Grinberg, Travis Scrimshaw
• Status changed from needs_review to positive_review

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

### comment:18 Changed 7 years ago by vbraun

• Branch changed from public/symmetric_functions/remove_adams_op to 5f785a2df1f0f3fc033fe7e5d933914fcf7d43e4
• Resolution set to fixed
• Status changed from positive_review to closed
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