Skew Schur Functions of Sums of Fat Staircases

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. If a diagram's skew Schur function is a linear combination of Schur functions of fat staircases, we call the diagram a sum of fat staircases. We prove a Schur-positivity result that is obtained each time we augment a sum of fat staircases with a skew diagram. We also determine conditions on which diagrams can be sums of fat staircases, including necessary and sufficient conditions in the special case when the diagram is a fat staircase skew a single row or column.

26 pages