  We define a fat staircase to be a Ferrers diagram corresponding to a
partition of the form $(n^{\alpha_n}, {n-1}^{\alpha_{n-1}},..., 1^{\alpha_1})$,
where $\alpha = (\alpha_1,...,\alpha_n)$ is a composition, or the $180^\circ$
rotation of such a diagram. We look at collections of skew diagrams consisting
of a fixed fat staircase augmented with all hooks of a given size. Among these
diagrams we determine precisely which pairs give a Schur-positive difference.
We extend this classification to collections of fat staircases augmented with
hook-complements.
