Symmetric-function involution ($\omega (e_r)=h_r,\omega (p_r)=(-1{)}^{r-1}{p}_r$)

The Fundamental theorem of symmetric polynomials gives freely generated elementary functions, so sending each $e_r$ to $h_r$ defines an algebra homomorphism. For $E(t)=\sum e_r{t}^r$ and $H(t)=\sum h_r{t}^r$, $E(-t)H(t)=1$. 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.