Editing Ring of symmetric functions (section)

=== 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.