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.