Differences of Skew Schur Functions of Staircases with Transposed Foundations

TL;DR

This paper investigates the Schur-positivity of differences of skew Schur functions derived from staircase diagrams with transposed foundations, focusing on cases where foundations are simple or hook-shaped.

Contribution

It introduces the concept of staircases with bad foundations and proves Schur-positivity for differences when foundations are one, two rows, or hooks.

Findings

Proves Schur-positivity for specific foundation shapes.

Defines staircases with transposed foundations.

Identifies cases where differences are Schur-positive.

Abstract

We consider the skew diagram $\Delta_n$, which is the $180^\circ$ rotation of the staircase diagram $\delta_n = (n,n-1,n-2,...,2,1)$. We create a staircase with bad foundation by augmenting $\Delta_n$ with another skew diagram, which we call the \textit{foundation}. We consider pairs of staircases with bad foundation whose foundations are transposes of one another. Among these pairs, we show that the difference of the corresponding skew Schur functions is Schur-positive in the case when one of the foundations consists of either a one or two row diagram, or a hook diagram.