On Integral Forms of Specht Modules Labelled by Hook Partitions

We investigate integral forms of simple modules of symmetric groups over fields of characteristic $0$ labelled by hook partitions. Building on work of Plesken and Craig, for every odd prime $p$, we give a set of representatives of the isomorphism classes of ${\mathbb{Z}}_p$-forms of the simple $$\mathbb{Q}$_p \mathfrak{S}_n$-modulelabelledbythepartition$(n-k,1^k)$,where$n\in$\mathbb{N}$$and$0\leq k\leq n-1$. We also settle the analogous question for $p=2$, assuming that $n\equiv ̸0(\mathrm{m}\mathrm{o}\mathrm{d}4)$ and $k\in \{2,n-3\}$. As a consequence this leads to a set of representatives of the isomorphism classes of $\mathbb{Z}$-forms of the simple $\mathbb{Q}{\mathfrak{S}}_n$-modules labelled by $(n-2,1^2)$ and $(3,1^{n-3})$, again assuming $n\equiv ̸0(\mathrm{m}\mathrm{o}\mathrm{d}4)$.