Stability of Kronecker products of irreducible characters of the symmetric group
\textit{F. D. Murnaghan} [Am. J. Math. 60, 761-784 (1938; Zbl 0018.29701)] showed that the decomposition of the Kronecker product \(\chi^{(n-a,\lambda_2,\dots)}\otimes \chi^{(n-b,\mu_2,\dots)}\) of two irreducible characters of the symmetric group depends only on \(\overline\lambda=(\lambda_2,\dots)\) and \(\overline\mu=(\mu_2,\dots)\) but not on \(n\). However, this result is only valid for arbitrary \(n\) if one applies modification rules (sometimes called ``slinky rules'') to disordered ``partitions'' appearing as indices of characters in the decomposition. Hence, if one denotes by \(c(\lambda,\mu,\nu)\) the coefficient of \(\chi^\nu\) in the expansion of \(\chi^{\lambda}\otimes \chi^{\mu}\), then \(c((n-a,\lambda_2,\dots),(n-b,\mu_2,\dots),(n-c,\nu_2,\dots))\) will be independent of \(n\) only for large \(n\), but may vary for small \(n\). The author provides a precise bound for this phenomenon, i.e., he gives an integer \(m\) so that \(c((n-a,\lambda_2,\dots),(n-b,\mu_2,\dots),(n-c,\nu_2,\dots))\) is independent of \(n\) for all \(n\geq m\). His bound is not optimal, though. The author's proof is completely combinatorial. It is based on manipulating Littlewood--Richardson multitableaux and rim hook tabloids, by means of which the coefficients \(c\) can be expressed. In particular, the proof does not require the truth of Murnaghan's result.
Kronecker product of characters of the symmetric group
scientific article; zbMATH DE number 1351195