Simple symmetric functions

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Sym = symmetrisation

• $m_\lambda(x) = Sym(x_1^{\lambda_1} ... x_\ell^{\lambda_\ell})$

• $e_\lambda = \prod_{i = 1 : \ell} e_{\lambda_i}$

where

• $e_n(x) = Sym(x_1...x_n)$

Generating function:

$$E(t, x) = \prod_{1 \ge 1} (1 + t x_i)$$

• $h_\lambda = \prod_{i = 1 : \ell} h_{\lambda_i}$

where

• $h_n(x) = \sum_{n_1 + ... + n_k = n} x^n$

generating function:

$$H(t, x) = \sum_{n \ge 0} t^n h_n(x) = \prod_{i \ge 1}{1 \over 1 - t x_i}.$$

• $H(t) E(-t) = 1$
• so $\sum_{r = 0 : n} (-1)^r h_{n - r} e_r = 0$, i.e. $e_n = h_1 e_{n - 1} - h_2 e_{n - 2} + ...$.
• The above relation shows that homomorphism $\omega$ defined by $\omega (e_n) = h_n$ is an involution.
• $s_{(n)} = h_n$ and $s_{(n)'} = e_n$.

Define $f_\lambda := \omega(m_\lambda)$, called the forgotten symmetric functions.

[{macdonald98}]: no simple direct description.

• $p_\lambda = \prod_{i = 1 : \ell} p_{\lambda_i}$

where

• $p_n(x) = Sym(x_1^n)$

Dirichlet generating function ([{sagan00}]):

$$\sum_{n \ge 1} p_n(x) {t^n \over n} = \log \prod_{i \ge 1} {1 \over 1 - x_i t} = \log H(t, x).$$

Generating function ([{macdonald98}]):

$$\begin{align} P(t) = \sum_{r \ge 1} p_r t^{r - 1} = H'(t) / H(t). P(-t) = E'(t) / E(t) \end{align}$$

so

$$\begin{align} n h_n = \sum_{r = 1 : n} p_r h_{n - r}\\ n e_n = \sum_{r = 1 : n} (-1)^{r - 1} p_r e_{n - r} \end{align}$$

The second one above is Newton's formula.

So

$$\omega(p_n) = (-1)^{n - 1} p_n, \qquad \omega(p_\lambda) = (-1)^{|\lambda| - l(\lambda)} p_\lambda =: \epsilon_\lambda p_\lambda.$$

References

• [macdonald98] Symmetric Functions and Hall Polynomials, I. MacDonald, 1998.
• [sagan00] The Symmetric Group, B.E. Sagan, 2000.