# S(e_n) = (-1)^n h_n, and a mistake in lecture

1. In class I explained that $S(e_{\lambda}) = (-1)^n h_{\lambda}$ for any partition $\lambda$ of $n$, and that this follows from the equation

$\sum_{k=0}^n (-1)^k h_k e_{n-k} = 0$

for $n \geq 1$. Can someone provide a proof of this?

(Hint: there are elegant formulas for the generating functions $\sum_n e_nt^n$ and $\sum_n h_nt^n$.)

2. I made a mistake in defining the inner product on Sym. I meant to decree that $m_{\lambda}$ and $h_{\lambda}$ are dual bases. (I wrote e instead of m). So we have:

$\langle m_{\lambda}, h_{\mu} \rangle = 1$ if $\lambda = \mu$, and $\langle m_{\lambda}, h_{\mu} \rangle = 0$ if $\lambda \neq \mu$.