On \(p\)-Kostka numbers and Young modules.
Let \(r\geq 1\). Let \(\Sigma_r\) denote the symmetric group on \(r\) letters. Let \(p\) be a prime dividing \(r!\). Let \(\lambda\) be a partition of \(r\). Let \(M^\lambda_p\) be the \(\mathbb{F}_p\Sigma_r\)-module obtained by induction of the trivial module \(\mathbb{F}_p\) from the Young subgroup \(\Sigma_\lambda\) to \(\Sigma_r\). The Specht module \(S^\lambda_p\) is a submodule of \(M^\lambda_p\). Let the Young module \(Y^\lambda_p\) be a direct summand of \(M^\lambda_p\) containing \(S^\lambda_p\). This defines \(Y^\lambda_p\) up to isomorphism. The module \(M^\lambda_p\) decomposes into a direct sum of Young modules. Given partitions \(\mu\) and \(\lambda\) of \(r\), we let the \(p\)-Kostka number \([M^\mu_p:Y^\lambda_p]\) be the multiplicity of \(Y^\lambda_p\) as a direct summand of \(M^\mu_p\). James showed that, using the dominance ordering, the \(p\)-Kostka numbers form a unipotent triangular matrix. Klyachko and Donkin gave a combinatorial criterion as to whether an entry in this matrix is nonzero. In this article, first, \(p\)-Kostka numbers are determined in the case of two-part partitions \(\mu\) and \(\lambda\). It turns out that they are \(\leq 1\). Second, let \(\lambda\) be a two-part partition and let \(Y^\lambda_0\) be a \(\mathbb{Z}_p\Sigma_\lambda\)-module reducing to \(Y^\lambda_p\). The corresponding ordinary character \(\text{ch\,}Y^\lambda_0\) is explicitly decomposed into irreducible characters. Third, the restricted Young module \(Y^\lambda_p\!\!\downarrow^{\Sigma_r}_{\Sigma_{r-1}}\) for a two-part partition \(\lambda\) is explicitly decomposed into a direct sum of Young modules. Here, multiplicities up to \(2\) occur. Fourth, as a conjecture it is stated that for general partitions \(\lambda\) and \(\mu\) of \(r\) with \(\lambda_1=\mu_1\), we have \([M^\mu_p:Y^\lambda_p]=[M^{\widetilde\mu}_p:Y^{\widetilde\lambda}_p]\), where \(\widetilde\mu=(\mu_2,\mu_3,\dots)\) is \(\mu\) with first row removed, and likewise \(\widetilde\lambda\). The author shows that this conjecture is true provided it holds in the particular case if \(\lambda_1<p\).
scientific article; zbMATH DE number 2183842