Some plethysm results related to Foulkes' conjecture

Steven Sivek

The electronic journal of combinatorics, Tome 13 (2006)

Résumé

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.

Détail

DOI : 10.37236/1050

Classification : 05E10