Ring of symmetric functions: Difference between revisions

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== Symmetric polynomials ==

{{main article | Symmetric polynomial }}

The study of symmetric functions is based on that of symmetric polynomials. In a [[polynomial ring]] in some finite set of indeterminates, a polynomial is called ''symmetric'' if it stays the same whenever the indeterminates are permuted in any way. More formally, there is an [[group action|action]] by [[ring homomorphism|ring automorphism]]s of the [[symmetric group]] ''S_n'' on the polynomial ring in ''n'' indeterminates, where a permutation acts on a polynomial by simultaneously substituting each of the indeterminates for another according to the permutation used. The [[Invariant (mathematics)#Unchanged under group action|invariants]] for this action form the subring of symmetric polynomials. If the indeterminates are ''X''₁,...,''X''_''n'', then examples of such symmetric polynomials are

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

Another construction of Λ_''R'' takes somewhat longer to describe, but better indicates the relationship with the rings ''R''[''X''₁,...,''X''_''n'']^{'''S'''_''n''} of symmetric polynomials in ''n'' indeterminates. For every ''n'' there is a surjective [[ring homomorphism]] ρ_''n'' from the analoguousanalogous ring ''R''[''X''₁,...,''X''_''n''+1]^{'''S'''_''n''+1} with one more indeterminate onto ''R''[''X''₁,...,''X''_''n'']^{'''S'''_''n''}, defined by setting the last indeterminate ''X''_''n''+1 to&nbsp;0. Although ρ_''n'' has a non-trivial kernel, the nonzero elements of that kernel have degree at least <math>n+1</math> (they are multiples of ''X''₁''X''₂...''X''_''n''+1). This means that the restriction of ρ_''n'' to elements of degree at most ''n'' is a bijective linear map, and ρ_''n''(''e''_''k''(''X''₁,...,''X''_''n''+1))&nbsp;=&nbsp;''e''_''k''(''X''₁,...,''X''_''n'') for all ''k''&nbsp;≤&nbsp;''n''. The inverse of this restriction can be extended uniquely to a ring homomorphism φ_''n'' from ''R''[''X''₁,...,''X''_''n'']^{'''S'''_''n''} to ''R''[''X''₁,...,''X''_''n''+1]^{'''S'''_''n''+1}, as follows for instance from the [[fundamental theorem of symmetric polynomials]]. Since the images φ_''n''(''e''_''k''(''X''₁,...,''X''_''n''))&nbsp;=&nbsp;''e''_''k''(''X''₁,...,''X''_''n''+1) for ''k''&nbsp;=&nbsp;1,...,''n'' are still [[algebraically independent]] over&nbsp;''R'', the homomorphism φ_''n'' is injective and can be viewed as a (somewhat unusual) inclusion of rings; applying φ_''n'' to a polynomial amounts to adding all monomials containing the new indeterminate obtained by symmetry from monomials already present. The ring Λ_''R'' is then the "union" ([[direct limit]]) of all these rings subject to these inclusions. Since all φ_''n'' are compatible with the grading by total degree of the rings involved, Λ_''R'' 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_''n'' without mentioning the injective morphisms φ_''n'': it constructs the homogeneous components of Λ_''R'' separately, and equips their direct sum with a ring structure using the ρ_''n''. It is also observed that the result can be described as an [[inverse limit]] in the 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''₁,...,''X''_''d'']^{'''S'''_''d''}. 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_''n'' 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.