Zonotopal algebra

A wealth of geometric and combinatorial properties of a given linear endomorphism X of IR 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 := rankX ≪ 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.