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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.
[[User:John Baez|John Baez]] ([[User talk:John Baez|talk]])
== Complete (homogeneous) symmetric functions ==
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