Editing Ring of symmetric functions (section)

=== Structural properties of Λ<sub>''R''</sub> ===
Important properties of Λ<sub>''R''</sub> include the following.

# The set of monomial symmetric functions parametrized by partitions form a basis of Λ<sub>''R''</sub> as a graded ''R''-[[module (mathematics)|module]], those parametrized by partitions of ''d'' being homogeneous of degree ''d''; the same is true for the set of Schur functions (also parametrized by partitions).
# 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&nbsp;''i'' for all ''i''&nbsp;>&nbsp;0, one isomorphism being the one that sends ''Y''<sub>''i''</sub> to ''e''<sub>''i''</sub>&nbsp;∈&nbsp;Λ<sub>''R''</sub> for every&nbsp;''i''.
# There is an [[Involution (mathematics)|involutory]] [[automorphism]] ω of Λ<sub>''R''</sub> that interchanges the elementary symmetric functions ''e''<sub>''i''</sub> and the complete homogeneous symmetric function ''h''<sub>''i''</sub> for all ''i''. It also sends each power sum symmetric function ''p''<sub>''i''</sub> to (−1)<sup>''i''−1</sup>''p''<sub>''i''</sub>, and it permutes the Schur functions among each other, interchanging ''s''<sub>λ</sub> and ''s''<sub>λ<sup>t</sup></sub> where λ<sup>t</sup> is the transpose partition of λ.

Property 2 is the essence of the [[fundamental theorem of symmetric polynomials]]. It immediately implies some other properties:
* The subring of Λ<sub>''R''</sub> generated by its elements of degree at most ''n'' is isomorphic to the ring of symmetric polynomials over ''R'' in ''n'' variables;
* The [[Hilbert–Poincaré series]] of Λ<sub>''R''</sub> is <math>\textstyle\prod_{i=1}^\infty\frac1{1-t^i}</math>, the [[generating function]] of the [[integer partition]]s (this also follows from property&nbsp;1);
* For every ''n''&nbsp;>&nbsp;0, the ''R''-module formed by the homogeneous part of Λ<sub>''R''</sub> of degree ''n'', modulo its intersection with the subring generated by its elements of degree strictly less than ''n'', is [[free module|free]] of rank&nbsp;1, and (the image of) ''e''<sub>''n''</sub> is a generator of this ''R''-module;
* For every family of symmetric functions (''f''<sub>''i''</sub>)<sub>''i''>0</sub> in which ''f''<sub>''i''</sub> is homogeneous of degree&nbsp;''i'' and gives a generator of the free ''R''-module of the previous point (for all ''i''), there is an alternative isomorphism of graded ''R''-algebras from ''R''[''Y''<sub>1</sub>,''Y''<sub>2</sub>, ...] as above to Λ<sub>''R''</sub> that sends ''Y''<sub>''i''</sub> to ''f''<sub>''i''</sub>; in other words, the family (''f''<sub>''i''</sub>)<sub>''i''>0</sub> forms a set of free polynomial generators of Λ<sub>''R''</sub>.

This final point applies in particular to the family (''h''<sub>''i''</sub>)<sub>''i''>0</sub> of complete homogeneous symmetric functions. 
If ''R'' contains the [[field (mathematics)|field]]&nbsp;<math>\mathbb Q</math> of [[rational number]]s, it applies also to the family (''p''<sub>''i''</sub>)<sub>''i''>0</sub> of power sum symmetric functions. This explains why the first ''n'' elements of each of these families define sets of symmetric polynomials in ''n'' variables that are free polynomial generators of that ring of symmetric polynomials.

The fact that the complete homogeneous symmetric functions form a set of free polynomial generators of Λ<sub>''R''</sub> already shows the existence of an automorphism&nbsp;ω sending the elementary symmetric functions to the complete homogeneous ones, as mentioned in property&nbsp;3. The fact that ω is an involution of Λ<sub>''R''</sub> follows from the symmetry between elementary and complete homogeneous symmetric functions expressed by the first set of relations given above.

The ring of symmetric functions Λ<sub>'''Z'''</sub> is the [[Exp ring]] of the integers '''Z'''. It is also a [[Λ-ring|lambda-ring]] in a natural fashion; in fact it is the universal lambda-ring in one generator.