Ring of symmetric functions: Difference between revisions

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where the <math>e_i</math> denote elementary symmetric polynomials; this formula is valid for all natural numbers ''n'', and the only notable dependency on it is that ''e''_''k''(''X''₁,…,''X''_''n'')&nbsp;=&nbsp;0 whenever ''n''&nbsp;&lt;&nbsp;''k''. One would like to write this as an identity

:<math>p_3=e_1^3-3e_2 e_1 + 3e_3</math>

that does not depend on ''n'' at all, and this can be done in the ring of symmetric polynomialsfunctions. In that ring there are elements ''e''_''k'' for all integers ''k''&nbsp;≥&nbsp;1, and any element of the ring can be given by a polynomial expression in the elements ''e''_''k''.

=== Definitions ===