Differences of Augmented Staircase Skew Schur Functions

TL;DR

This paper investigates the Schur-positivity of differences between augmented fat staircase skew diagrams, providing a precise classification of pairs that yield positive differences and extending the analysis to hook-complements.

Contribution

It introduces a detailed classification of Schur-positive differences for augmented fat staircase skew diagrams and extends the results to hook-complement cases.

Findings

Identifies pairs of augmented fat staircase diagrams with Schur-positive differences.

Provides a complete classification for diagrams augmented with hooks of a fixed size.

Extends the classification to include hook-complement augmented diagrams.

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.