Some Plethysm Results related to Foulkes' Conjecture

Abstract

We provide several classes of examples to show that Stanley's plethysm conjecture and a reformulation by Pylyavskyy, both concerning the ranks of certain matrices $K^{\lambda}$ associated with Young diagrams $\lambda$, are in general false. We also provide bounds on the rank of $K^{\lambda}$ by which it may be possible to show that the approach of Black and List to Foulkes' conjecture does not work in general. Finally, since Black and List's work concerns $K^{\lambda}$ for rectangular shapes $\lambda$, we suggest a constructive way to prove that $K^{\lambda}$ does not have full rank when $\lambda$ is a large rectangle.