# Simple symmetric functions

Sym = symmetrisation

## Monomial

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

## Elementary

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

where

• $$e_n(x) = Sym(x_1...x_n)$$

Generating function:

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

## Complete homogenous

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

where

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

generating function:

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

## Relation between $$e$$ and $$h$$

• $$H(t) E(-t) = 1$$
• 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 $$s_{(n)'} = e_n$$.

## Forgotten

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

[{macdonald98}]: no simple direct description.

## Power sum

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

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, , 1998.
• [sagan00] The Symmetric Group, , 2000.