= Symmetric-function involution
{title2=$\omega(e_r)=h_r,\quad\omega(p_r)=(-1)^{r-1}p_r$}
The <Fundamental theorem of symmetric polynomials> gives freely generated elementary functions, so sending each $e_r$ to $h_r$ defines an <algebra homomorphism>. For $E(t)=\sum e_rt^r$ and $H(t)=\sum h_rt^r$, $E(-t)H(t)=1$. Applying the homomorphism shows $\omega(H(t))=E(t)$, hence $\omega^2=1$. Taking logarithms of $H$ gives the displayed action on power sums. It preserves the <Hall inner product of symmetric functions> because each power-sum <basis> vector acquires only a sign.
Back to article page
