Zonotopal algebra

By Olga Holtz, Amos Ron. 2007

56 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/c9efa1740818f4fde047d04ce1abff3938cec029

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Computational Theory and Mathematics

In Discrete Mathematics and Combinatorics

Citations in the corpus

Sagbi Bases of Cox-Nagata Rings (2008, methodology): “This formula is derived by applying [18, Theorem 4.2] to the rank d matroid on v 1 +· · ·+ v m elements which is obtained from M(C) by duplicating the elements 1 , . . . , m respectively v 1 , . . . , v m times.”
A Tutte polynomial for toric arrangements (2009, background): “The relations between zonotopes and Dahmen-Micchelli spaces has been studied intensively in recent years, giving rise to many algebraic and combinatorial constructions (see in particular, [9], [13], [10], [2], [14], [18]).”
Arithmetic matroids, the Tutte polynomial and toric arrangements (2011, background): “More recently, Holtz and Ron in [16], and Ardila and Postnikov in [1] introduced various algebraic structures capturing a rich description of these objects.”
Combinatorics and geometry of power ideals (2008, methodology): “• (Dahmen-Micchelli [6], De Concini-Procesi [8], Holtz-Ron [11]) Given a finite set X = {a1, .”