Differences of Augmented Staircase Skew Schur Functions

Abstract

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.