Ring of symmetric functions: Difference between revisions

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The following are fundamental examples of symmetric functions.

* The '''monomial symmetric functions''' ''m''_&alpha;. Suppose &alpha;&nbsp;=&nbsp;(&alpha;₁,&alpha;₂,&hellip;) determinedis bya sequence of non-negative integers, only finitely many of which are non-zero. Then we can consider the [[monomial]] defined by &alpha;: ''X''^&alpha; (where &alpha;&nbsp;=&nbsp;(''X''₁^&alpha;₁,''X''₂^&alpha;₂,''X''₃^&alpha;₃&hellip;). isThen a''m''_&alpha; finiteis sequencethe ofsymmetric naturalfunction numbers);determined by ''mX''<subsup>&alpha;</subsup>, isi.e. the sum of all monomials obtained by symmetry from ''X''^&alpha; by symmetry. For a formal definition, consider such sequences to be infinite by appending zeroes (which does not alter the monomial), and define the relation "&beta;~"&alpha; betweento such sequencesmean that expressesthe thatsequence one&beta; is a permutation of the othersequence &alpha; thenand set

::<math>m_\alpha=\sum\nolimits_{\beta\sim\alpha}X^\beta.</math>

:This symmetric function corresponds to the [[monomial symmetric polynomial]] ''m''_&alpha;(''X''₁,&hellip;,''X''_''n'') for any ''n'' large enough to have the monomial ''X''^&alpha;. The distinct monomial symmetric functions are parametrized by the [[integer partition]]s (each ''m''_&alpha; has a unique representative monomial ''X''^&lambda; with the parts </sup>&lambda;_''i'' in weakly decreasing order). Since any symmetric function containing any of the monomials of some ''m''_&alpha; 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 &Lambda;_''R'' as ''R''-[[module (mathematics)|module]].

* The '''elementary symmetric functions''' ''e''_''k'', for any natural number ''k''; one has ''e''_''k''&nbsp;=&nbsp;''m''_&alpha; where <math>X^\alpha=\Pi_{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,&hellip;,''X''_''n'') for any ''n''&nbsp;&ge;&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,&hellip;,''X''_''n'')&nbsp;=&nbsp;''X''₁^''k''+&hellip;+''X''_''n''^''k'' for any ''n''&nbsp;&ge;&nbsp;1.