Some plethysm results related to Foulkes' conjecture
type: Item
instance of: scholarly article
title: Some plethysm results related to Foulkes' conjecture
zbMATH Open document ID: 1084.05075
author: Steven Sivek
published in: The Electronic Journal of Combinatorics
publication date: 2006-03-22T00:00:00Z
full work available at URL: https://eudml.org/doc/125741
full work available at URL: v13i1r24.html
review text: Summary: We provide several classes of examples to show that Stanley's ple\-thysm conjecture [\textit{P. R. Stanley}, in: Mathematics: frontiers and perspectives, 295--316 (2000; Zbl 0955.05111)] and a reformulation by \textit{P. Pylyavskyy} [Electron. J. Comb. 11, Research paper 8, electronic only (2004; Zbl 1060.05099)], 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; see \textit{S. C. Black} and \textit{R. J. List} [Eur. J. Comb. 10, 111--112 (1989; Zbl 0668.20017)]. 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.
Mathematics Subject Classification ID: 05E10
zbMATH DE Number: 5014321
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5014321