## 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.

## Exercise

Using one of Newton’s identities, express as a polynomial in:

## Self-Duality

Recall that the involution swaps for Looking at the expressions of in terms of and the following is clear.

Proposition. For each , we have

**Proof**

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:

## Summary

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.