Cyclotomic polytopes and growth series of cyclotomic lattices

Abstract

The coordination sequence of a lattice $\L$ encodes the word-length function with respect to $M$, a set that generates $\L$ as a monoid. We investigate the coordination sequence of the cyclotomic lattice $\L = \Z[\zeta_m]$, where $\zeta_m$ is a primitive $m\th$ root of unity and where $M$ is the set of all $m\th$ roots of unity. We prove several conjectures by Parker regarding the structure of the rational generating function of the coordination sequence; this structure depends on the prime factorization of $m$. Our methods are based on unimodular triangulations of the $m\th$ cyclotomic polytope, the convex hull of the $m$ roots of unity in $\R^{\phi(m)}$, with respect to a canonically chosen basis of $\L$.