Zeta Functions of Lattices of the Symmetric Group

The symmetric group ${\mathfrak{S}}_{n+1}$ of degree $n+1$ admits an $n$-dimensional irreducible $\mathbf{Q}{\mathfrak{S}}_n$-module $V$ corresponding to the hook partition $(2,1^{n-1})$. By the work of Craig and Plesken we know that there are $\sigma (n+1)$ many isomorphism classes of $\mathbf Z \mathfrak S_{n+1}$-latticeswhicharerationallyequivalentto$V$,where$\sigma$ denotes the divisor counting function. In the present paper we explicitly compute the Solomon zeta function of these lattices. As an application we obtain the Solomon zeta function of the $\mathbf{Z}{\mathfrak{S}}_{n+1}$-lattice defined by the Specht basis.