= Hodge star eigenvalues on graded differential forms
{c}
{title2=$\operatorname{Spec}(*)=\{\pm1\}\ (n\text{ odd}),\quad\{\pm1,\pm i\}\ (n\text{ even}>0)$}
Use the complex-linear <Hodge star operator> on the direct sum of all form degrees. Its square on degree $r$ is $(-1)^{r(n-r)}$. In positive odd dimension this is always one; in positive even dimension its sign is positive on even-degree forms and negative on odd-degree forms. Complementary basis wedges provide both roots of each sign, so the displayed <eigenvalues> all occur. On a single degree the <Hodge star operator> is an endomorphism only in middle degree; on middle degree $m$ in dimension $2m$ its possible <eigenvalues> are $\pm1$ for even $m$ and $\pm i$ for odd $m$.
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