Ring of symmetric functions: Difference between revisions

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The following are fundamental examples of symmetric functions.

* The '''monomial symmetric functions''' ''m''_α. Suppose α = (α₁,α₂,...) 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''^α = ''X''₁^α₁''X''₂^α₂''X''₃^α₃... . Then ''m''_α is the symmetric function determined by ''X''^α, i.e. the sum of all monomials obtained from ''X''^α 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''_α(''X''₁,...,''X''_''n'') for any ''n'' large enough to have the monomial ''X''^α. The distinct monomial symmetric functions are parametrized by the [[integer partition]]s (each ''m''_α has a unique representative monomial ''X''^&thinsp;λ with the parts λ_''i'' in weakly decreasing order). Since any symmetric function containing any of the monomials of some ''m''_α 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_''R'' as an ''R''-[[module (mathematics)|module]].

* The '''elementary symmetric functions''' ''e''_''k'', for any natural number ''k''; one has ''e''_''k''&nbsp;=&nbsp;''m''_α 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''_''k''(''X''₁,...,''X''_''n'') for any ''n''&nbsp;≥&nbsp;''k''.

* The '''power sum symmetric functions''' ''p''_''k'', for any positive integer ''k''; one has ''p''_''k''&nbsp;=&nbsp;''m''_(''k''), the monomial symmetric function for the monomial ''X''₁^''k''. This symmetric function corresponds to the [[power sum symmetric polynomial]] ''p''_''k''(''X''₁,...,''X''_''n'')&nbsp;=&nbsp;''X''₁^''k'' + ... + ''X''_''n''^''k'' for any ''n''&nbsp;≥&nbsp;1.

* The '''complete homogeneous symmetric functions''' ''h''_''k'', for any natural number ''k''; ''h''_''k'' is the sum of all monomial symmetric functions ''m''_α 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''_''k''(''X''₁,...,''X''_''n'') for any ''n''&nbsp;≥&nbsp;''k''.

* 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''^&thinsp;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 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.

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=== Generating functions ===

The first definition of Λ_''R'' 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_''R'', these expressions involve operations taking place in ''R''<nowiki>[[</nowiki>''X''₁,''X''₂,...;''t''&hairsp;<nowiki>]]</nowiki> but outside its subring Λ_''R''<nowiki>[[</nowiki>''t''&hairsp;<nowiki>]]</nowiki>, so they are meaningful only if symmetric functions are viewed as formal power series in indeterminates ''X''_''i''. We shall write "(''X'')" after the symmetric functions to stress this interpretation.

The generating function for the elementary symmetric functions is

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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_''k''>0&nbsp;''p''_''k''(''X'')&hairsp;''t''^{&hairsp;''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>).