On the structure of Specht modules in the principal block of $F\Sigma_{3p}$

Abstract

Let $F$ be a field of characteristic $p$ at least 5. We study the Loewy structures of Specht modules in the principal block of $F\Sigma_{3p}$. We show that a Specht module in the block has Loewy length at most 4 and composition length at most 14. Furthermore, we classify which Specht modules have Loewy length 1, 2, 3, or 4, produce a Specht module having 14 composition factors, describe the second radical layer and the socle of the reducible Specht modules, and prove that if a Specht module corresponds to a partition that is $p$-regular and $p$-restricted then the head of the Specht module does not extend the socle.