The Structure of the Complex of Maximal Lattice Free Bodies for a Matrix of Size (n + 1) × n

Abstract

The complex of maximal lattice free bodies associated with a well behaved matrix A of size (n+1)×n is generated by a finite set of simplicies, K ⁰(A), of the form 0, h¹, …, h ^k∼, with k≤n, and their lattice translates. The simplicies in K ⁰(A) are selected so that the plane a ⁰ x=0, with a ⁰ the first row of A, passes through the vertex 0. The collection of simplicies h ¹, …, h ^k is denoted by Top. Various properties of Top are demonstrated, including the fact that no two interior faces of Top are lattice translates of each other. Moreover, if g is a generator of the cone generated by the set of neighbors h with a ⁰ h>0, then the set of simplicies of Top which contain g is the union of linear intervals of simplicies with special features. These features lead to an algorithm for calculating the simplicies in K ₀ (A) as a ⁰ varies and the plane a ⁰ x=0 passes through the generator g.

I am grateful to Sasha Barvinok, Dave Bayer, Anders Björner, Raymond Hemmecke, Roger Howe, Bjarke Roune, David Shallcross, Bernd Sturmfels, and Kevin Woods for their intellectual company over the course of many years. I would also like to acknowledge my great debt to Imre Báarány, Ravi Kannan, Niels Lauritzen, Rekha Thomas and Zaifu Yang for their thoughtful, patient and constructive reading of the material in this paper.