Ring of Symmetric Functions - The Ring of Symmetric Functions

The Ring of Symmetric Functions

Most relations between symmetric polynomials do not depend on the number n of indeterminates, other than that some polynomials in the relation might require n to be large enough in order to be defined. For instance the Newton's identity for the third power sum polynomial leads to

where the denote elementary symmetric polynomials; this formula is valid for all natural numbers n, and the only notable dependency on it is that e_k(X₁,…,X_n) = 0 whenever n < k. One would like to write this as an identity p₃ = e₁3 − 3e₂e₁ + 3e₃ that does not depend on n at all, and this can be done in the ring of symmetric polynomials. In that ring there are elements e_k for all integers k ≥ 1, and an arbitrary element can be given by a polynomial expression in them.