Some plethysm results related to Foulkes' conjecture
The electronic journal of combinatorics, Tome 13 (2006)
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.
@article{10_37236_1050,
author = {Steven Sivek},
title = {Some plethysm results related to {Foulkes'} conjecture},
journal = {The electronic journal of combinatorics},
year = {2006},
volume = {13},
doi = {10.37236/1050},
zbl = {1084.05075},
url = {http://geodesic.mathdoc.fr/articles/10.37236/1050/}
}
Steven Sivek. Some plethysm results related to Foulkes' conjecture. The electronic journal of combinatorics, Tome 13 (2006). doi: 10.37236/1050
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