A Principle Relating Symmetric Polynomials and Symmetric Functions
For any symmetric function P, the corresponding symmetric polynomials in n indeterminates for any natural number n may be designated by P(X1,…,Xn). The second definition of the ring of symmetric functions implies the following fundamental principle:
- If P and Q are symmetric functions of degree d, then one has the identity of symmetric functions if and only one has the identity P(X1,…,Xd) = Q(X1,…,Xd) of symmetric polynomials in d indeterminates. In this case one has in fact P(X1,…,Xn) = Q(X1,…,Xn) for any number n of indeterminates.
This is because one can always reduce the number of variables by substituting zero for some variables, and one can increase the number of variables by applying the homomorphisms φn; the definition of those homomorphisms assures that φn(P(X1,…,Xn)) = P(X1,…,Xn+1) (and similarly for Q) whenever n ≥ d. See a proof of Newton's identities for an effective application of this principle.
Read more about this topic: Ring Of Symmetric Functions, The Ring of Symmetric Functions
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