Zeta Functions of Lattices of the Symmetric Group

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

On Lattices of Integral Group Algebras and Solomon Zeta Functions

“…, where p is any prime, have been determined by the second author in [6]. Here we shall investigate S (n−r,1 r ) Zp , for all r ∈ {1, .…”

“…The concrete computation of zeta functions of ZG-lattices is in general a rather difficult problem, and not too much is known in this direction. The case where L is the regular ZG-lattice has been studied most intensively; for a list of known results see [6]. In [6], the second author determined the zeta functions ζ ZSn (L, s),…”

“…The case where L is the regular ZG-lattice has been studied most intensively; for a list of known results see [6]. In [6], the second author determined the zeta functions ζ ZSn (L, s),…”

Computing zeta functions of table algebra orders using the Bushnell-Reiner integral approach

“…Bushnell and Reiner developed a method for calculating the Solomon zeta function for a Λ-lattice, where Λ is an integral order in a semisimple algebra over a p-adic field that makes use of ideas from Tate's thesis [5]. Since then this approach has been used by Wittmann to compute submodule zeta functions for the ZC p -lattice (ZC p ) n [18] and by Hofmann to compute the submodule zeta function for the Z-lattice corresponding to a ZS n+1 -module affording the irreducible character related to the hook partition (2, 1 n−1 ) [14]. The examples in this paper illustrate how to apply their technique to calculate, for any relevant prime p, the zeta function of Z p B where B is the Z-basis consisting of the standard basis of a commutative integral table algebra, a situation which includes the cases where B is a finite abelian group or the set of adjacency matrices of a commutative association scheme as special cases.…”