Zeta Functions of Lattices of the Symmetric Group

TL;DR

This paper explicitly computes the Solomon zeta functions of certain lattices of the symmetric group, revealing detailed algebraic properties and providing new explicit formulas for these zeta functions.

Contribution

It provides the first explicit computation of Solomon zeta functions for lattices associated with the symmetric group’s irreducible modules.

Findings

Explicit formulas for Solomon zeta functions of these lattices

Identification of the zeta function for the Specht basis lattice

Connection between zeta functions and lattice classification

Abstract

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 $\mathbf Z \mathfrak S_{n+1}$-lattices which are rationally equivalent to $V$, 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.