Cyclotomic polytopes and growth series of cyclotomic lattices

Abstract

The coordination sequence of a lattice script captial L encodes the word-length function with respect to M, a set that generates script captial L as a monoid. We investigate the coordination sequence of the cyclotomic lattice script captial L = ℤ [ζm], where ζm is a primitive mth 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 mth cyclotomic polytope, the convex hull of the m roots of unity in ℝφ(m), with respect to a canonically chosen basis of script captial L. © International Press 2006.