Talk:Ring of symmetric functions: Difference between revisions

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== Overlap ==

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:Agreed – I’ve moved the article to “[[Ring of symmetric functions]]” and narrowed its scope to that, while making a separate “[[Symmetric functions]]” article for general properties.

:—Nils von Barth ([[User:Nbarth|nbarth]]) ([[User talk:Nbarth|talk]]) 17:46, 21 April 2009 (UTC)

== Ring of functions ==

There's this article, and a [[ring of polynomial functions]] but no [[ring of functions]]. Can someone create that? That makes it hard to define what an [[operator product algebra]] is ... (its like an [[operator product expansion]] but more formally defined.)[[Special:Contributions/67.198.37.16|67.198.37.16]] ([[User talk:67.198.37.16|talk]]) 18:56, 3 September 2015 (UTC)

== Unclear statement ==

In the first way to define the ring of symmetric functions, "'''As a ring of formal power series'''", the second condition (Condition 2.) is that the degrees of the monomials allowed in one element must be bounded.

The explanation for this condition is given as follows:

"''Note that because of the second condition, power series are used here only to allow infinitely many terms of a fixed degree, rather than to sum terms of all possible degrees. Allowing this is necessary because an element that contains for instance a term X₁ should also contain a term X_i for every i > 1 in order to be symmetric.''"

I do not understand why Condition 2. is "necessary", as the second sentence claims. Yes, it is clear that "''an element that contains for instance a term X₁ should also contain a term X_i for every i > 1 in order to be symmetric''" — but what does this have to do with Condition 2. ???

Obviously, the subring of formal power series in infinitely many indeterminates X_i defined solely by Condition 1. — which requires that they are unchanged by the action of the permutation group S('''ℕ'''₀) of the nonnegative integers '''ℕ'''₀ on the indices — makes perfect sense. So this defines a subring that does not require Condition 2.

So: Can someone who is knowledgeable in this subject please explain what goal is accomplished by also requiring Condition 2. ???

And: There must be a name for the subring that satisfies Condition 1. alone. What is its name? In my opinion this subring should also be discussed here — at least a little bit! [[User:Daqu|Daqu]] ([[User talk:Daqu|talk]]) 01:52, 29 February 2016 (UTC)

: Condition 2 is necessary to define the usual concept of 'symmetric function'! If we leave it out, a lot of important theorems are no longer true, e.g. number 2 in the list of 'basic properties:

::: # Λ_''R'' is [[isomorphic]] as a graded ''R''-algebra to a polynomial ring ''R''[''Y''₁,''Y''₂, ...] in infinitely many variables, where ''Y''_''i'' is given degree ''i'' for all ''i'' > 0, one isomorphism being the one that sends ''Y''_''i'' to ''e''_''i'' ∈ Λ_''R'' for every ''i''.

: The point is that if we drop condition 2 we are allowing new elements in our ring, like ''e''_''0'' + ''e''_''1'' + ''e''_''2'' + <math>\cdots</math>

: "There must be a name for the subring that satisfies Condition 1. alone." I don't know a name for it, and the only thing I know to say about it is that it's larger than the ring of symmetric functions. - [[User:John Baez|John Baez]] ([[User talk:John Baez|talk]])

== Complete (homogeneous) symmetric functions ==

I think here and elsewhere we should change the term "[[complete homogeneous symmetric function]]" to "complete symmetric function" because:

1) This is the term Mac Donald uses in the important reference ''Symmetric Functions and Hall Polynomials''

2) Yes, these symmetric functions are homogeneous, but so are the power sum symmetric functions and elementary symmetric functions, and this page does not put the word "homogeneous" in their names.

Is there a reason we need to keep the word "homogeneous"?

[[User:John Baez|John Baez]] ([[User talk:John Baez|talk]]) 14:29, 2 August 2024 (UTC)