Editing Ring of symmetric functions (section)

== The ring of symmetric functions ==
Most relations between symmetric polynomials do not depend on the number ''n'' of indeterminates, other than that some polynomials in the relation might require ''n'' to be large enough in order to be defined. For instance the [[Newton's identities|Newton's identity]] for the third power sum polynomial ''p<sub>3</sub>'' leads to
:<math>p_3(X_1,\ldots,X_n)=e_1(X_1,\ldots,X_n)^3-3e_2(X_1,\ldots,X_n)e_1(X_1,\ldots,X_n)+3e_3(X_1,\ldots,X_n),</math>
where the <math>e_i</math> denote elementary symmetric polynomials; this formula is valid for all [[natural number]]s ''n'', and the only notable dependency on it is that ''e''<sub>''k''</sub>(''X''<sub>1</sub>,...,''X''<sub>''n''</sub>)&nbsp;=&nbsp;0 whenever ''n''&nbsp;<&nbsp;''k''. One would like to write this as an identity 
:<math>p_3=e_1^3-3e_2 e_1 + 3e_3</math>
that does not depend on ''n'' at all, and this can be done in the ring of symmetric functions. In that ring there are nonzero elements ''e''<sub>''k''</sub> for all [[integer]]s ''k''&nbsp;≥&nbsp;1, and any element of the ring can be given by a polynomial expression in the elements ''e''<sub>''k''</sub>.

=== Definitions ===

A '''ring of symmetric functions''' can be defined over any [[commutative ring]] ''R'', and will be denoted Λ<sub>''R''</sub>; the basic case is for ''R''&nbsp;=&nbsp;'''Z'''. The ring Λ<sub>''R''</sub> is in fact a [[graded ring|graded]] ''R''-[[Algebra over a ring|algebra]]. There are two main constructions for it; the first one given below can be found in (Stanley, 1999), and the second is essentially the one given in (Macdonald, 1979).

==== As a ring of formal power series ====

The easiest (though somewhat heavy) construction starts with the ring of [[Formal power series#Power series in several variables|formal power series]] <math>R[[X_1,X_2,...]]</math> over ''R'' in infinitely ([[countably infinite|countably]]) many indeterminates; the elements of this [[power series]] ring are formal infinite sums of terms, each of which consists of a coefficient from ''R'' multiplied by a [[monomial]], where each monomial is a product of finitely many finite powers of indeterminates. One defines Λ<sub>''R''</sub> as its subring consisting of those power series ''S'' that satisfy 
#''S'' is invariant under any permutation of the indeterminates, and
#the [[degree of a polynomial|degrees]] of the monomials occurring in ''S'' are bounded.
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''&nbsp;>&nbsp;1 in order to be symmetric. Unlike the whole power series ring, the subring Λ<sub>''R''</sub> is graded by the total degree of monomials: due to condition&nbsp;2, every element of Λ<sub>''R''</sub> is a finite sum of [[Homogeneous polynomial|homogeneous]] elements of Λ<sub>''R''</sub> (which are themselves infinite sums of terms of equal degree). For every ''k''&nbsp;≥&nbsp;0, the element ''e''<sub>''k''</sub>&nbsp;∈&nbsp;Λ<sub>''R''</sub> is defined as the formal sum of all products of ''k'' distinct indeterminates, which is clearly homogeneous of degree ''k''.

==== As an algebraic limit ====

Another construction of Λ<sub>''R''</sub> takes somewhat longer to describe, but better indicates the relationship with the rings ''R''[''X''<sub>1</sub>,...,''X''<sub>''n''</sub>]<sup>'''S'''<sub>''n''</sub></sup> of symmetric polynomials in ''n'' indeterminates. For every ''n'' there is a [[surjective]] [[ring homomorphism]] ''ρ''<sub>''n''</sub> from the analogous ring ''R''[''X''<sub>1</sub>,...,''X''<sub>''n''+1</sub>]<sup>'''S'''<sub>''n''+1</sub></sup> with one more indeterminate onto ''R''[''X''<sub>1</sub>,...,''X''<sub>''n''</sub>]<sup>'''S'''<sub>''n''</sub></sup>, defined by setting the last indeterminate ''X''<sub>''n''+1</sub> to&nbsp;0. Although ''ρ''<sub>''n''</sub> has a non-trivial [[kernel (algebra)|kernel]], the nonzero elements of that kernel have degree at least <math>n+1</math> (they are multiples of ''X''<sub>1</sub>''X''<sub>2</sub>...''X''<sub>''n''+1</sub>). This means that the restriction of ''ρ''<sub>''n''</sub> to elements of degree at most ''n'' is a [[bijective]] [[linear map]], and ''ρ''<sub>''n''</sub>(''e''<sub>''k''</sub>(''X''<sub>1</sub>,...,''X''<sub>''n''+1</sub>))&nbsp;=&nbsp;''e''<sub>''k''</sub>(''X''<sub>1</sub>,...,''X''<sub>''n''</sub>) for all ''k''&nbsp;≤&nbsp;''n''. The inverse of this restriction can be extended uniquely to a ring homomorphism ''φ''<sub>''n''</sub> from ''R''[''X''<sub>1</sub>,...,''X''<sub>''n''</sub>]<sup>'''S'''<sub>''n''</sub></sup> to ''R''[''X''<sub>1</sub>,...,''X''<sub>''n''+1</sub>]<sup>'''S'''<sub>''n''+1</sub></sup>, as follows for instance from the [[fundamental theorem of symmetric polynomials]]. Since the images ''φ''<sub>''n''</sub>(''e''<sub>''k''</sub>(''X''<sub>1</sub>,...,''X''<sub>''n''</sub>))&nbsp;=&nbsp;''e''<sub>''k''</sub>(''X''<sub>1</sub>,...,''X''<sub>''n''+1</sub>) for ''k''&nbsp;=&nbsp;1,...,''n'' are still [[algebraically independent]] over&nbsp;''R'', the homomorphism ''φ''<sub>''n''</sub> is [[injective]] and can be viewed as a (somewhat unusual) inclusion of rings; applying ''φ''<sub>''n''</sub> to a polynomial amounts to adding all monomials containing the new indeterminate obtained by symmetry from monomials already present. The ring Λ<sub>''R''</sub> is then the "union" ([[direct limit]]) of all these rings subject to these inclusions. Since all ''φ''<sub>''n''</sub> are compatible with the grading by total degree of the rings involved, Λ<sub>''R''</sub> obtains the structure of a graded ring.

This construction differs slightly from the one in (Macdonald, 1979). That construction only uses the surjective morphisms ''ρ''<sub>''n''</sub> without mentioning the injective morphisms ''φ''<sub>''n''</sub>: it constructs the homogeneous components of Λ<sub>''R''</sub> separately, and equips their [[direct sum]] with a ring structure using the ''ρ''<sub>''n''</sub>. It is also observed that the result can be described as an [[inverse limit]] in the [[category (mathematics)|category]] of ''graded'' rings. That description however somewhat obscures an important property typical for a ''direct'' limit of injective morphisms, namely that every individual element (symmetric function) is already faithfully represented in some object used in the limit construction, here a ring ''R''[''X''<sub>1</sub>,...,''X''<sub>''d''</sub>]<sup>'''S'''<sub>''d''</sub></sup>. It suffices to take for ''d'' the degree of the symmetric function, since the part in degree ''d'' of that ring is mapped isomorphically to rings with more indeterminates by ''φ''<sub>''n''</sub> for all ''n''&nbsp;≥&nbsp;''d''. This implies that for studying relations between individual elements, there is no fundamental difference between symmetric polynomials and symmetric functions.

=== Defining individual symmetric functions ===

The name "symmetric function" for elements of Λ<sub>''R''</sub> is a [[misnomer]]: in neither construction are the elements [[function (mathematics)|functions]], and in fact, unlike symmetric polynomials, no function of independent variables can be associated to such elements (for instance ''e''<sub>1</sub> would be the sum of all infinitely many variables, which is not defined unless restrictions are imposed on the variables). However the name is traditional and well established; it can be found both in (Macdonald, 1979), which says (footnote on p.&nbsp;12)
<blockquote>The elements of Λ (unlike those of Λ<sub>''n''</sub>) are no longer polynomials: they are formal infinite sums of monomials. We have therefore reverted to the older terminology of symmetric functions.</blockquote>
(here Λ<sub>''n''</sub> denotes the ring of symmetric polynomials in ''n'' indeterminates), and also in (Stanley, 1999).

To define a symmetric function one must either indicate directly a power series as in the first construction, or give a symmetric polynomial in ''n'' indeterminates for every natural number ''n'' in a way compatible with the second construction. An expression in an unspecified number of indeterminates may do both, for instance
:<math>e_2=\sum_{i<j}X_iX_j\,</math>
can be taken as the definition of an elementary symmetric function if the number of indeterminates is infinite, or as the definition of an elementary symmetric polynomial in any finite number of indeterminates. Symmetric polynomials for the same symmetric function should be compatible with the homomorphisms ''ρ''<sub>''n''</sub> (decreasing the number of indeterminates is obtained by setting some of them to zero, so that the coefficients of any monomial in the remaining indeterminates is unchanged), and their degree should remain bounded. (An example of a family of symmetric polynomials that fails both conditions is <math>\textstyle\prod_{i=1}^nX_i</math>; the family <math>\textstyle\prod_{i=1}^n(X_i+1)</math> fails only the second condition.) Any symmetric polynomial in ''n'' indeterminates can be used to construct a compatible family of symmetric polynomials, using the homomorphisms ''ρ''<sub>''i''</sub> for ''i''&nbsp;<&nbsp;''n'' to decrease the number of indeterminates, and ''φ''<sub>''i''</sub> for ''i''&nbsp;≥&nbsp;''n'' to increase the number of indeterminates (which amounts to adding all monomials in new indeterminates obtained by symmetry from monomials already present).

The following are fundamental examples of symmetric functions.
* The '''monomial symmetric functions''' ''m''<sub>α</sub>. Suppose α&nbsp;=&nbsp;(α<sub>1</sub>,α<sub>2</sub>,...) is a sequence of non-negative integers, only finitely many of which are non-zero. Then we can consider the [[monomial]] defined by α: ''X''<sup>α</sup> = ''X''<sub>1</sub><sup>α<sub>1</sub></sup>''X''<sub>2</sub><sup>α<sub>2</sub></sup>''X''<sub>3</sub><sup>α<sub>3</sub></sup>.... Then ''m''<sub>α</sub> is the symmetric function determined by ''X''<sup>α</sup>, i.e. the sum of all monomials obtained from ''X''<sup>α</sup> by symmetry. For a formal definition, define β ~ α to mean that the sequence β is a permutation of the sequence α and set
::<math>m_\alpha=\sum\nolimits_{\beta\sim\alpha}X^\beta.</math>
:This symmetric function corresponds to the [[monomial symmetric polynomial]] ''m''<sub>α</sub>(''X''<sub>1</sub>,...,''X''<sub>''n''</sub>) for any ''n'' large enough to have the monomial ''X''<sup>α</sup>. The distinct monomial symmetric functions are parametrized by the [[integer partition]]s (each ''m''<sub>α</sub> has a unique representative monomial ''X''<sup>λ</sup> with the parts λ<sub>''i''</sub> in weakly decreasing order). Since any symmetric function containing any of the monomials of some ''m''<sub>α</sub> must contain all of them with the same coefficient, each symmetric function can be written as an ''R''-linear combination of monomial symmetric functions, and the distinct monomial symmetric functions therefore form a basis of Λ<sub>''R''</sub> as an ''R''-[[module (mathematics)|module]].
* The '''elementary symmetric functions''' ''e''<sub>''k''</sub>, for any natural number ''k''; one has ''e''<sub>''k''</sub>&nbsp;=&nbsp;''m''<sub>α</sub> where <math>\textstyle 
X^\alpha=\prod_{i=1}^kX_i</math>. As a power series, this is the sum of all distinct products of ''k'' distinct indeterminates. This symmetric function corresponds to the [[elementary symmetric polynomial]] ''e''<sub>''k''</sub>(''X''<sub>1</sub>,...,''X''<sub>''n''</sub>) for any ''n''&nbsp;≥&nbsp;''k''.
* The '''power sum symmetric functions''' ''p''<sub>''k''</sub>, for any positive integer ''k''; one has ''p''<sub>''k''</sub>&nbsp;=&nbsp;''m''<sub>(''k'')</sub>, the monomial symmetric function for the monomial ''X''<sub>1</sub><sup>''k''</sup>. This symmetric function corresponds to the [[power sum symmetric polynomial]] ''p''<sub>''k''</sub>(''X''<sub>1</sub>,...,''X''<sub>''n''</sub>)&nbsp;=&nbsp;''X''<sub>1</sub><sup>''k''</sup> + ... + ''X''<sub>''n''</sub><sup>''k''</sup> for any ''n''&nbsp;≥&nbsp;1.
* The '''complete homogeneous symmetric functions''' ''h''<sub>''k''</sub>, for any natural number ''k''; ''h''<sub>''k''</sub> is the sum of all monomial symmetric functions ''m''<sub>α</sub> where α is a [[integer partition|partition]] of&nbsp;''k''. As a power series, this is the sum of ''all'' monomials of degree ''k'', which is what motivates its name. This symmetric function corresponds to the [[complete homogeneous symmetric polynomial]] ''h''<sub>''k''</sub>(''X''<sub>1</sub>,...,''X''<sub>''n''</sub>) for any ''n''&nbsp;≥&nbsp;''k''.
* The '''Schur functions''' ''s''<sub>λ</sub> for any partition λ, which corresponds to the [[Schur polynomial]] ''s''<sub>λ</sub>(''X''<sub>1</sub>,...,''X''<sub>''n''</sub>) for any ''n'' large enough to have the monomial ''X''<sup>λ</sup>.

There is no power sum symmetric function ''p''<sub>0</sub>: although it is possible (and in some contexts natural) to define <math>\textstyle p_0(X_1,\ldots,X_n)=\sum_{i=1}^nX_i^0=n</math> as a symmetric ''polynomial'' in ''n'' variables, these values are not compatible with the morphisms ''ρ''<sub>''n''</sub>. The "discriminant" <math>\textstyle(\prod_{i<j}(X_i-X_j))^2</math> is another example of an expression giving a symmetric polynomial for all ''n'', but not defining any symmetric function. The expressions defining [[Schur polynomial]]s as a quotient of alternating polynomials are somewhat similar to that for the discriminant, but the polynomials ''s''<sub>λ</sub>(''X''<sub>1</sub>,...,''X''<sub>''n''</sub>) turn out to be compatible for varying ''n'', and therefore do define a symmetric function.

=== A principle relating symmetric polynomials and symmetric functions ===

For any symmetric function ''P'', the corresponding symmetric polynomials in ''n'' indeterminates for any natural number ''n'' may be designated by ''P''(''X''<sub>1</sub>,...,''X''<sub>''n''</sub>). The second definition of the ring of symmetric functions implies the following fundamental principle:

:If ''P'' and ''Q'' are symmetric functions of degree ''d'', then one has the identity <math>P=Q</math> of symmetric functions [[if and only if]] one has the identity ''P''(''X''<sub>1</sub>,...,''X''<sub>''d''</sub>)&nbsp;=&nbsp;''Q''(''X''<sub>1</sub>,...,''X''<sub>''d''</sub>) of symmetric polynomials in ''d'' indeterminates. In this case one has in fact ''P''(''X''<sub>1</sub>,...,''X''<sub>''n''</sub>)&nbsp;=&nbsp;''Q''(''X''<sub>1</sub>,...,''X''<sub>''n''</sub>) for ''any'' number ''n'' of indeterminates.

This is because one can always reduce the number of variables by substituting zero for some variables, and one can increase the number of variables by applying the homomorphisms ''φ''<sub>''n''</sub>; the definition of those homomorphisms assures that ''φ''<sub>''n''</sub>(''P''(''X''<sub>1</sub>,...,''X''<sub>''n''</sub>))&nbsp;=&nbsp;''P''(''X''<sub>1</sub>,...,''X''<sub>''n''+1</sub>) (and similarly for ''Q'') whenever ''n''&nbsp;≥&nbsp;''d''. See [[Newton's identities#Derivation of the identities|a proof of Newton's identities]] for an effective application of this principle.