Ring of symmetric functions: Difference between revisions

Revision as of 03:00, 18 August 2019

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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 morphisms ρ_''n'' (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>\Pi_textstyle\prod_{i=1}^nX_i</math>; the family <math>\Pi_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 morphisms ρ_''i'' for ''i''&nbsp;<&nbsp;''n'' to decrease the number of indeterminates, and φ_''i'' 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.

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::<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''^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 ''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=\Pi_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''^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)=\Sigma_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.

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