ZETA FUNCTIONS OF LATTICES OF THE SYMMETRIC GROUP

The symmetric group n+1 of degree n + 1 admits an n-dimensional irreducible Qn-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 Zn+1-lattices which are rationally equivalent to V, where sigma denotes the divisor counting function. In the present article, we explicitly compute the Solomon zeta function of these lattices. As an application we obtain the Solomon zeta function of the Zn+1-lattice defined by the Specht basis.