Zonotopal algebra

Abstract

A wealth of geometric and combinatorial properties of a given linear endomorphism X of R N is captured in the study of its associated zonotope Z ( X ) , and, by duality, its associated hyperplane arrangement H ( X ) . This well-known line of study is particularly interesting in case n : = rank X ≪ N . We enhance this study to an algebraic level, and associate X with three algebraic structures, referred herein as external, central, and internal. Each algebraic structure is given in terms of a pair of homogeneous polynomial ideals in n variables that are dual to each other: one encodes properties of the arrangement H ( X ) , while the other encodes by duality properties of the zonotope Z ( X ) . The algebraic structures are defined purely in terms of the combinatorial structure of X, but are subsequently proved to be equally obtainable by applying suitable algebro-analytic operations to either of Z ( X ) or H ( X ) . The theory is universal in the sense that it requires no assumptions on the map X (the only exception being that the algebro-analytic operations on Z ( X ) yield sought-for results only in case X is unimodular), and provides new tools that can be used in enumerative combinatorics, graph theory, representation theory, polytope geometry, and approximation theory.