Ring of symmetric functions: Difference between revisions

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{{main | 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 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

: <math>X_1+X_2+\cdots+X_n, \, </math>

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* The '''Schur functions''' ''s''_λ for any partition λ, which corresponds to the [[Schur polynomial]] ''s''_λ(''X''₁,...,''X''_''n'') for any ''n'' large enough to have the monomial ''X''^λ.

There is no power sum symmetric function ''p''₀: 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''_''n''. 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 polynomial|alternating polynomials]] are somewhat similar to that for the discriminant, but the polynomials ''s''_λ(''X''₁,...,''X''_''n'') turn out to be compatible for varying ''n'', and therefore do define a symmetric function.

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

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Property 2 is the essence of the [[fundamental theorem of symmetric polynomials]]. It immediately implies some other properties:

* The subring of Λ_''R'' 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_''R'' is <math>\textstyle\prod_{i=1}^\infty\frac1{1-t^i}</math>, the [[Partition (number theory)#Generating function|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_''R'' 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''_''n'' is a generator of this ''R''-module;

* For every family of symmetric functions (''f''_''i'')_''i''>0 in which ''f''_''i'' 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''₁,''Y''₂, ...] as above to Λ_''R'' that sends ''Y''_''i'' to ''f''_''i''; in other words, the family (''f''_''i'')_''i''>0 forms a set of free polynomial generators of Λ_''R''.

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((Macdonald, 1979) defines ''P''(''t'') as Σ_''k''>0&nbsp;''p''_''k''(''X'')''t''^''k''−1, 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>).

== Specializations ==