Editing Ring of symmetric functions (section)

== Properties of the ring of symmetric functions ==

=== Identities ===

The ring of symmetric functions is a convenient tool for writing identities between symmetric polynomials that are independent of the number of indeterminates: in Λ<sub>''R''</sub> there is no such number, yet by the above principle any identity in Λ<sub>''R''</sub> automatically gives identities the rings of symmetric polynomials over ''R'' in any number of indeterminates. Some fundamental identities are
:<math>\sum_{i=0}^k(-1)^ie_ih_{k-i}=0=\sum_{i=0}^k(-1)^ih_ie_{k-i}\quad\mbox{for all }k>0,</math>
which shows a symmetry between elementary and complete homogeneous symmetric functions; these relations are explained under [[complete homogeneous symmetric polynomial]].
:<math>ke_k=\sum_{i=1}^k(-1)^{i-1}p_ie_{k-i}\quad\mbox{for all }k\geq0,</math>
the [[Newton identities]], which also have a variant for complete homogeneous symmetric functions:
:<math>kh_k=\sum_{i=1}^kp_ih_{k-i}\quad\mbox{for all }k\geq0.</math>

=== 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.

=== Generating functions ===

The first definition of Λ<sub>''R''</sub> as a subring of <math>R[[X_1, X_2, ...]]</math> allows the [[generating function]]s of several sequences of symmetric functions to be elegantly expressed. Contrary to the relations mentioned earlier, which are internal to Λ<sub>''R''</sub>, these expressions involve operations taking place in ''R''<nowiki>[[</nowiki>''X''<sub>1</sub>,''X''<sub>2</sub>,...;''t''<nowiki>]]</nowiki> but outside its subring Λ<sub>''R''</sub><nowiki>[[</nowiki>''t''<nowiki>]]</nowiki>, so they are meaningful only if symmetric functions are viewed as formal power series in indeterminates ''X''<sub>''i''</sub>. We shall write "(''X'')" after the symmetric functions to stress this interpretation.

The generating function for the elementary symmetric functions is
:<math>E(t) = \sum_{k \geq 0} e_k(X)t^k = \prod_{i=1}^\infty (1+X_it).</math>
Similarly one has for complete homogeneous symmetric functions
:<math>H(t) = \sum_{k \geq 0} h_k(X)t^k = \prod_{i=1}^\infty \left(\sum_{k \geq 0} (X_it)^k\right) = \prod_{i=1}^\infty \frac1{1-X_it}.</math>
The obvious fact that <math>E(-t)H(t) = 1 = E(t)H(-t)</math> explains the symmetry between elementary and complete homogeneous symmetric functions.
The generating function for the power sum symmetric functions can be expressed as
:<math>P(t) = \sum_{k>0} p_k(X)t^k = \sum_{k>0}\sum_{i=1}^\infty (X_it)^k = \sum_{i=1}^\infty\frac{X_it}{1-X_it} = \frac{tE'(-t)}{E(-t)} = \frac{tH'(t)}{H(t)}</math>
((Macdonald, 1979) defines ''P''(''t'') as Σ<sub>''k''>0</sub>&nbsp;''p''<sub>''k''</sub>(''X'')''t''<sup>''k''−1</sup>, and its expressions therefore lack a factor ''t'' with respect to those given here). The two final expressions, involving the [[formal derivative]]s of the generating functions ''E''(''t'') and ''H''(''t''), imply Newton's identities and their variants for the complete homogeneous symmetric functions. These expressions are sometimes written as
:<math>P(t) = -t\frac d{dt}\log(E(-t)) = t\frac d{dt}\log(H(t)),</math>
which amounts to the same, but requires that ''R'' contain the rational numbers, so that the logarithm of power series with constant term&nbsp;1 is defined (by <math>\textstyle\log(1-tS) = -\sum_{i>0} \frac1i(tS)^i</math>).