Zonotopal algebra

Abstract

A wealth of geometric and combinatorial properties of a given linear endomorphism $X$ of ${\mathbb{R}}^N$ is captured in the study of its associated zonotope $Z(X)$, and, by duality, its associated hyperplane arrangement $\mathcal{H}(X)$. This well-known line of study is particularly interesting in case $n\eqbd\rank X \ll N$. We enhance this study to an algebraic level, and associate $X$ with three algebraic structures, referred herein as {\it 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 $\mathcal{H}(X)$, 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 $\mathcal{H}(X)$. 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.

44 pages; updated to reflect referees' remarks and the developments in the area since the article first appeared on the arXiv

References in corpus (3)

• Toric Hyperkahler Varieties
• Products of Linear Forms and Tutte Polynomials
• Zonotopal algebra

Cited by in corpus (9)

• Zonotopal algebra
• Hierarchical zonotopal power ideals
• Zonotopal algebra and forward exchange matroids
• Interpolation, box splines, and lattice points in zonotopes
• Fourientations and the Tutte Polynomial
• Splines, lattice points, and arithmetic matroids
• Generic and special constructions of pure O-sequences
• Lattice points in polytopes, box splines, and Todd operators
• Combinatorics and geometry of power ideals