On signed $p$-Kostka matrices

Abstract

We show that the signed $p$-Kostka numbers depend just on $p$-Kostka numbers and the multiplicities of projective indecomposable modules in certain signed Young permutation modules. We then examine the signed $p$-Kostka number $k_{(\alpha |\beta ),(\lambda |p\mu )}$ in the case when $|\beta |=p|\mu |$. This allows us to explicitly describe the multiplicities of direct summands of a signed Young permutation module lying in the principal block of $F{\mathfrak{S}}_{mp}$ in terms of the $p$-Kostka numbers.