Skew Schur Functions of Sums of Fat Staircases

TL;DR

This paper introduces the concept of fat staircases in Ferrers diagrams and studies when skew Schur functions can be expressed as linear combinations of such diagrams, proving Schur-positivity under certain augmentations.

Contribution

It defines fat staircases and characterizes diagrams whose skew Schur functions are sums of fat staircases, including necessary and sufficient conditions in specific cases.

Findings

Proves Schur-positivity when augmenting sums of fat staircases with skew diagrams.

Provides necessary and sufficient conditions for diagrams to be sums of fat staircases.

Characterizes fat staircase skew diagrams with single row or column.

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.