  Postnikov constructed a cellular decomposition of the totally nonnegative
Grassmannians. The poset of cells can be described (in particular) via
Grassmann necklaces. We study certain quiver Grassmannians for the cyclic
quiver admitting a cellular decomposition, whose cells are naturally labeled by
Grassmann necklaces. We show that the posets of cells coincide with the
reversed cell posets of the cellular decomposition of the totally nonnegative
Grassmannians. We investigate algebro-geometric and combinatorial properties of
these quiver Grassmannians. In particular, we describe the irreducible
components, study the action of the automorphism groups of the underlying
representations and describe the moment graphs. We also construct a resolution
of singularities for each irreducible component; the resolutions are defined as
quiver Grassmannians for an extended cyclic quiver.
