Let $V$ be a vector space with some extra structure (maybe to be general, an object of an abelian tensor category?), where we can form tensor products and exterior powers $\Lambda^i V$ and symmetric powers $\text{Sym}^i V$. I believe the following always holds for $n > 0$: $$ \sum_{i=0}^n (-1)^i\Lambda^i V \otimes \text{Sym}^{n-i}V = 0$$ where I take the sum as a "virtual" object; another way to say this is just that $$ \sum_{i=0}^{[n/2]} \Lambda^{2i} V \otimes \text{Sym}^{n-2i}V = \sum_{i=0}^{[(n-1)/2]} \Lambda^{2i+1} V \otimes \text{Sym}^{n-2i-1}V.$$

For example, if $V$ is a representation of a finite group, you can use character theory to reduce to the analogous identity for the elementary symmetric and complete homogeneous polynomials (see Wikipedia) applied to the eigenvalues of a group element. But it'd be nice to have a higher-level proof that applied more generally.