Plane partitions and characters of the symmetric group
type: Item
instance of: scholarly article
title: Plane partitions and characters of the symmetric group
zbMATH Open document ID: 0951.05098
DOI: 10.1023/A:1008795704190
author: Ernesto Vallejo
published in: Journal of Algebraic Combinatorics
publication date: 2000-12-11T00:00:00Z
review text: In general, the Kronecker product of two irreducible characters of the symmetric group is a reducible character. Irreducible characters of the symmetric group, \(S(n)\), of degree \(n\), can be labelled by partitions of \(n\); denote the character corresponding to the partition \(\lambda\) by \(\chi^{\lambda}\). Then there is a simple formula, going back at least to \textit{F. D. Murnaghan} [The analysis of the Kronecker product of irreducible representations of the symmetric group, Am. J. Math. 60, No. 3, 761-784 (1938; Zbl 0019.25103)], giving the multiplicity \(c(\lambda,\mu,\nu)\) of \(\chi^{\nu}\) in \(\chi^{\lambda}\otimes \chi^{\mu}\) as \(1/{n!}\sum_{\sigma\in S(n)} \chi^{\lambda}(\sigma)\chi^{\mu} (\sigma)\chi^{\nu}(\sigma)\). In this paper, the authors show that the existence of certain plane partitions which are minimal (in the sense of the dominance order) yields summands which are minimal in the dominance order; they also give a description of the multiplicity of minimal summands. The proof of these results uses the combinatorics of tableaux and plane partitions.
Mathematics Subject Classification ID: 05E10
Mathematics Subject Classification ID: 20C30
zbMATH DE Number: 1428367
zbMATH Keywords: tableau
zbMATH Keywords: symmetric group
zbMATH Keywords: Littelwood-Richardson multitableaux
zbMATH Keywords: Kronecker product
zbMATH Keywords: dominance order of partitions
zbMATH Keywords: character
reviewed by: Bethany R. Marsh
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1428367