The coefficients of lowest and next-highest degree of a linear operator's characteristic polynomial are its determinant and trace. These have well-known geometric interpretations. But what about its intermediate coefficients?

For a linear operator $f : V \to V$, we have the beautiful formula

$$\chi(f) = det(f - t) = \sum_{i=0}^n (-1)^i\ tr(\wedge^{n-i}(f))\ t^i,$$

where $\wedge^{p}(f)$ is the map induced by $f$ on grade $p$ of $V$'s exterior algebra.

While this formula is rarely mentioned (at least I haven't seen it in any of the standard textbooks), it is not too surprising if you have a good grasp of exterior algebra. It presents $\chi(f)$ as a generating function for the exterior traces of $f$.

My question is whether these traces have a simple geometric interpretation on par with $tr$ and $det$.