Differences of Skew Schur Functions of Staircases with Transposed Foundations

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.

33 pages