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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<sub>''R''</sub> is [[isomorphic]] as a graded ''R''-algebra to a polynomial ring ''R''[''Y''<sub>1</sub>,''Y''<sub>2</sub>, ...] in infinitely many variables, where ''Y''<sub>''i''</sub> is given degree ''i'' for all ''i'' > 0, one isomorphism being the one that sends ''Y''<sub>''i''</sub> to ''e''<sub>''i''</sub> ∈ Λ<sub>''R''</sub> for every ''i''.
: The point is that if we drop condition 2 we are allowing new elements in our ring, like ''e''<sub>''0''</sub> + ''e''<sub>''1''</sub> + ''e''<sub>''2''</sub> + <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)
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