Nonconvexity of the set of hypergraph degree sequences

It is well known that the set of possible degree sequences for a graph on $n$ vertices is the intersection of a lattice and a convex polytope. We show that the set of possible degree sequences for a $k$-uniform hypergraph on $n$ vertices is not the intersection of a lattice and a convex polytope for $k \geq 3$ and $n \geq k+13$. We also show an analogous nonconvexity result for the set of degree sequences of $k$-partite $k$-uniform hypergraphs and the generalized notion of $\lambda$-balanced $k$-uniform hypergraphs.Comment: 5 page

Relation spaces of hyperplane arrangements and modules defined by graphs of fiber zonotopes

We study the exactness of certain combinatorially defined complexes which generalize the Orlik-Solomon algebra of a geometric lattice. The main results pertain to complex reflection arrangements and their restrictions. In particular, we consider the corresponding relation complexes and give a simple proof of the $n$-formality of these hyperplane arrangements. As an application, we are able to bound the Castelnouvo-Mumford regularity of certain modules over polynomial rings associated to Coxeter arrangements (real reflection arrangements) and their restrictions. The modules in question are defined using the relation complex of the Coxeter arrangement and fiber polytopes of the dual Coxeter zonotope. They generalize the algebra of piecewise polynomial functions on the original arrangement

A Brief Survey on Lattice Zonotopes

Zonotopes are a rich and fascinating family of polytopes, with connections to many areas of mathematics. In this article we provide a brief survey of classical and recent results related to lattice zonotopes. Our emphasis is on connections to combinatorics, both in the sense of enumeration (e.g. Ehrhart theory) and combinatorial structures (e.g. graphs and permutations)

Zonotopal algebra

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 ${\cal H}(X)$. 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 ${\cal H}(X)$, 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 ${\cal H}(X)$. 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.Comment: 44 pages; updated to reflect referees' remarks and the developments in the area since the article first appeared on the arXi

Primitive Zonotopes

We introduce and study a family of polytopes which can be seen as a generalization of the permutahedron of type $B_d$. We highlight connections with the largest possible diameter of the convex hull of a set of points in dimension $d$ whose coordinates are integers between $0$ and $k$, and with the computational complexity of multicriteria matroid optimization.Comment: The title was slightly modified, and the determination of the computational complexity of multicriteria matroid optimization was adde