Differences of Augmented Staircase Skew Schur Functions

We define a fat staircase to be a Ferrers diagram corresponding to a partition of the form $(n^{α_n}, {n-1}^{α_{n-1}},..., 1^{α_1})$, where $α= (α_1,...,α_n)$ 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.

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