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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 X<sub>1</sub> should also contain a term X<sub>i</sub> 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 X<sub>1</sub> should also contain a term X<sub>i</sub> 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<sub>i</sub> defined solely by Condition 1. — which requires that they are unchanged by the action of the permutation group S('''ℕ'''<sub>0</sub>) of the nonnegative integers '''ℕ'''<sub>0</sub> 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)
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