  We introduce immanant varieties, associated to simple characters of a finite
group. They include well-studied classes of varieties, as Segre embeddings,
Grassmannians and certain other classes of Chow varieties. For a
one-dimensional character $\chi$, we define $\chi$-matroids by a maximality
property. For trivial characters, by exploring the combinatorics of incidence
stratifications, we provide a set of generators for the Chow vector spaces of
the corresponding immanant varieties.
