Polynomials and Representations IV

Power Sum Polynomials

The power sum polynomial is defined as follows:

p_k := x_1^k + x_2^k + \ldots + x_n^k \in \Lambda_n^{(k)}.

In this case, we do not define p_0, although it seems natural to set p_0 = n. As before, for a partition \lambda, define:

\displaystyle p_\lambda:= p_{\lambda_1} p_{\lambda_2} \ldots p_{\lambda_l}, \qquad l = l(\lambda).

Note that we must have l = l(\lambda) above since we have not defined p_0. For example if \lambda = (2, 1), then p_\lambda = \left(\sum_{i=1}^n x_i^2\right) \left(\sum_{i=1}^n x_i\right).

Their generating function is given by:

\displaystyle P(t) := p_1 + p_2 t + p_3 t^2 + \ldots= \sum_{j=1}^n \frac {x_j}{1 - x_j t}.

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Newton’s Identities (I)

Recall that the generating function for e_i is E(t) = \prod_{i=1}^n (1 + x_i t). Thus we have the following relation:

\displaystyle \log E(t) = \sum_{j=1}^n \log (1 + x_j t)\stackrel{\frac d {dt}}{\implies} \frac {E'(t)}{E(t)} = \sum_{j=1}^n \frac {x_j}{1 + x_j t} = P(-t).

From E'(t) = E(t) P(-t) we take the coefficient of t^{k-1} on both sides and obtain Newton’s identities:

k e_k = e_{k-1} p_1 - e_{k-2} p_2 + \ldots + (-1)^{k-1} e_0 p_k.

for all k\ge 1, where e_k = 0 for all k>n. With that, each p_k can be written as a polynomial in e_1, e_2, \ldots, e_n with integer coefficients. E.g.

p_1 = e_1,\quad p_2 = e_1^2 - 2e_2,\quad p_3 = e_1^3 - 3e_1 e_2 + 3e_3, \ldots

Conversely, we can also express each e_k as a polynomial in p_1, p_2, \ldots, p_n with rational coefficients. Thus we have:

\displaystyle \Lambda_n \otimes_{\mathbb{Z}} \mathbb{Q} =\mathbb{Q}[p_1, p_2, \ldots, p_n],

where the RHS is a free \mathbb{Q}-algebra in p_1, \ldots, p_n. In particular, \Lambda_n^{(d)}\otimes_{\mathbb{Z}} \mathbb{Q} has basis given by p_\lambda for all \lambda \vdash d, \lambda_1 \le n.

Newton’s Identities (II)

There is another form of Newton’s identities which relate p_k with the complete symmetric polynomials h_i. Recall that the generating function for h_i is H(t) = \prod_{i=1}^n \frac 1 {(1 - x_i t)}. Hence:

\displaystyle\log H(t) = -\sum_{j=1}^n \log(1 - x_j t) \implies \frac{H'(t)}{H(t)}= \sum_{j=1}^n \frac {x_j}{1 - x_j t} = P(t).

From H'(t) = P(t)H(t), taking the coefficient of t^{k-1} gives us another form of Newton’s identities:

k h_k = h_{k-1} p_1 + h_{k-2} p_2 + \ldots + h_0 p_k.

for all k\ge 1. From this, we can express p_k as a polynomial in h_1, \ldots, h_n with integer coefficients. E.g.

p_1 = h_1, \quad p_2 = 2h_2 - h_1^2,\quad p_3 = 3h_3 - 3h_2 h_1 + h_1^3.

As above, we can also express each h_k as a polynomial in p_1, \ldots, p _n with rational coefficients.

Exercise

Using one of Newton’s identities, express p_4 = x^4 + y^4 + z^4 as a polynomial in:

\begin{aligned} p_1 &= x+y+z,\\ p_2 &= x^2 + y^2 + z^2, \\ p_3 &= x^3 + y^3 + z^3.\end{aligned}

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Self-Duality

Recall that the involution \omega : \Lambda_n \to\Lambda_n swaps e_i \leftrightarrow h_i for 1\le i \le n. Looking at the expressions of p_n in terms of e_i and h_i, the following is clear.

Proposition. For each 1\le k \le n, we have \omega(p_k) = (-1)^{k-1}p_k.

Proof

Apply induction on k; when k=1 we have p_1 = e_1 = h_1 so this is clear. Suppose 1<k\le n. Apply \omega to the first Newton’s identity; for each i\le n we have \omega(e_i) = h_i and so:

k h_k = h_{k-1} \omega(p_1) - h_{k-2} \omega(p_2) + \ldots + (-1)^{k-1} h_0 \omega(p_k).

By induction hypothesis \omega(p_i) = (-1)^{i-1}p_i for 1 \le i \le k-1, so we get:

k h_k = h_{k-1} p_1 + h_{k-2} p_2 + \ldots + h_1 p_{k-1} + (-1)^{k-1} h_0 \omega(p_k).

Now the second Newton’s identity gives \omega(p_k) = (-1)^{k-1} p_k as desired. ♦

warningThe above is no longer true for k>n. For example, let n=2. We get:

\begin{aligned} p_1 = e_1\ &\implies\ \omega(p_1) = h_1 = p_1, \\ p_2 = e_1^2 - 2e_2\ &\implies\ \omega(p_2) = h_1^2 - 2h_2 = -p_2,\\ p_3 = e_1^3 - 3e_1 e_2 \ &\implies\ \omega(p_3) = h_1^3 - 3h_1 h_2 \ne \pm p_3.\end{aligned}

Summary

We have seen the following symmetric polynomials:

polynomials_relationsIn 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 p_\lambda when expressed as a linear sum of m_\mu. It turns out this has implications in the representation theory of the symmetric group S_n. We will (hopefully) get to talk about this beautiful result in a later article.

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