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Line 99: This final point applies in particular to the family (''h''<sub>''i''</sub>)<sub>''i''>0</sub> of complete homogeneous symmetric functions. If ''R'' contains the field ~~'''~~<math>\mathbb Q~~'''~~</math> of [[rational number]]s, it applies also to the family (''p''<sub>''i''</sub>)<sub>''i''>0</sub> of power sum symmetric functions. This explains why the first ''n'' elements of each of these families define sets of symmetric polynomials in ''n'' variables that are free polynomial generators of that ring of symmetric polynomials. The fact that the complete homogeneous symmetric functions form a set of free polynomial generators of Λ<sub>''R''</sub> already shows the existence of an automorphism ω sending the elementary symmetric functions to the complete homogeneous ones, as mentioned in property 3. The fact that ω is an involution of Λ<sub>''R''</sub> follows from the symmetry between elementary and complete homogeneous symmetric functions expressed by the first set of relations given above.

Ring of symmetric functions: Difference between revisions