A family of \(\mathbb{Z}\) bases for the ring of \(S_ n\) characters and applications to the decomposition of Kronecker products
The author constructs new \(\mathbb{Z}\) bases for the ring of characters of the symmetric groups. He uses these bases to continue his investigation on Kronecker products [J. Algebra 154, No. 1, 125-140 (1993)]. Fix two integers \(k, l\) in \(\mathbb{N}\). Given a partition \(\lambda = (\lambda_ 1, \lambda_ 2, \ldots)\) of an integer \(n\), let \(\lambda_{k,l} = (\lambda_ 1, \ldots, \lambda_ k, \widetilde{\lambda}_{k+1},\widetilde{\lambda}_{k+2},\ldots)\) where \(\widetilde{\lambda}_ j = \min(1,\lambda_ j)\). Let \(\lambda^{k,l} = \lambda/\lambda_{k,l}\). \(\lambda_{k,l}\) and \(\lambda^{k,l}\) are two partitions of the integers \(n_ 1\) and \(n_ 2\) such that \(n = n_ 1 + n_ 2\). Define a character of the symmetric group \(S_ n\) as follows: \(\Pi_{k,l}\lambda = \text{Ind}_{S_{n_ 1}\times S_{n_ 2}}^{S_ n} \lambda_{k,l} \times \lambda^{k,l}\), with the convention that the irreducible character indexed by a partition \(\alpha\) is denoted by \(\alpha\). The author shows that when \(\lambda\) runs through all the partitions of \(n\), \(\{\Pi_{k,l}\lambda\}\) is a \(\mathbb{Z}\) basis of the \(S_ n\) characters. For this he explicitly constructs the change of bases matrix and its inverse. He finally uses that result to give a recursive formula for the structure constants of the Kronecker product, and he gives a sufficient condition for these constants to be zero. Like many other algorithms, this recursive formula relies (heavily) on the Littlewood- Richardson rule.
scientific article; zbMATH DE number 683267