Plane partitions and characters of the symmetric group
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.
scientific article; zbMATH DE number 1428367
scientific article