# Generalized Foulkes modules and maximal and minimal constituents of plethysms of Schur functions

Paget, Rowena E., Wildon, Mark (2018) Generalized Foulkes modules and maximal and minimal constituents of plethysms of Schur functions. Proceedings of the London Mathematical Society, . ISSN 0024-6115. (doi:10.1112/plms.12210)

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http://dx.doi.org/10.1112/plms.12210

## Abstract

This paper proves a combinatorial rule giving all maximal and minimal partitions $\lambda$ such that the Schur function $s_\lambda$ appears in a plethysm of two arbitrary Schur functions. Determining the decomposition of these plethysms has been identified by Stanley as a key open problem in algebraic combinatorics. As corollaries we prove three conjectures of Agaoka on the partitions labeling the lexicographically greatest and least Schur functions appearing in an arbitrary plethysm. We also show that the multiplicity of the Schur function labelled by the lexicographically least constituent may be arbitrarily large. The proof is carried out in the symmetric group and gives an explicit non-zero homomorphism corresponding to each maximal or minimal partition.

Item Type: Article 10.1112/plms.12210 Q Science > QA Mathematics (inc Computing science) > QA165 Combinatorics Q Science > QA Mathematics (inc Computing science) > QA171 Representation theory Faculties > Sciences > School of Mathematics Statistics and Actuarial Science > Pure Mathematics Rowena E Paget 22 Aug 2016 10:52 UTC 29 May 2019 17:44 UTC https://kar.kent.ac.uk/id/eprint/56915 (The current URI for this page, for reference purposes)