Power Sum Polynomials
The power sum polynomial is defined as follows:
In this case, we do not define , although it seems natural to set As before, for a partition define:
Note that we must have above since we have not defined For example if , then
Their generating function is given by:
Newton’s Identities (I)
Recall that the generating function for is . Thus we have the following relation:
From we take the coefficient of on both sides and obtain Newton’s identities:
for all , where for all k>n. With that, each can be written as a polynomial in with integer coefficients. E.g.
Conversely, we can also express each as a polynomial in with rational coefficients. Thus we have:
where the RHS is a free -algebra in In particular, has basis given by for all
Newton’s Identities (II)
There is another form of Newton’s identities which relate with the complete symmetric polynomials Recall that the generating function for is Hence:
From , taking the coefficient of gives us another form of Newton’s identities:
for all . From this, we can express as a polynomial in with integer coefficients. E.g.
As above, we can also express each as a polynomial in with rational coefficients.
Using one of Newton’s identities, express as a polynomial in:
Recall that the involution swaps for Looking at the expressions of in terms of and the following is clear.
Proposition. For each , we have
Apply induction on k; when k=1 we have so this is clear. Suppose Apply to the first Newton’s identity; for each we have and so:
By induction hypothesis for so we get:
Now the second Newton’s identity gives as desired. ♦
The above is no longer true for k>n. For example, let n=2. We get:
We have seen the following symmetric polynomials:
In the above diagram a → b means that b can be expressed as a polynomial in a with integer coefficients; the same holds for dotted arrows but now the polynomial may have rational coefficients.
One wonders if there is a combinatorial interpretation of the coefficients of when expressed as a linear sum of It turns out this has implications in the representation theory of the symmetric group We will (hopefully) get to talk about this beautiful result in a later article.