arXiv subjects

Christos A. Athanasiadis

Publications and source records attributed to Christos A. Athanasiadis.

45 records · Page 3Linked to original sources

arXiv2007

Shellability and higher Cohen-Macaulay connectivity of generalized cluster complexes

Let $Φ$ be a finite root system of rank $n$ and let $m$ be a nonnegative integer. The generalized cluster complex $Δ^m (Φ)$ was introduced by S. Fomin and N. Reading. It was conjectured by these authors that $Δ^m (Φ)$ is shellable and by V. Reiner that it is $(m+1)$-Cohen-Macaulay, in the sense of Baclawski. These statements are proved in this paper. Analogous statements are shown to hold for the positive part $Δ^m_+ (Φ)$ of $Δ^m (Φ)$. An explicit homotopy equivalence is given between $Δ^m_+ (Φ)$ and the poset of generalized noncrossing partitions, associated to the pair $(Φ, m)$ by D. Armstrong.

math.CO↗

arXiv2006

A combinatorial reciprocity theorem for hyperplane arrangements

Given a nonnegative integer $m$ and a finite collection ${\mathcal A}$ of linear forms on ${\mathbb Q}^d$, the arrangement of affine hyperplanes in ${\mathbb Q}^d$ defined by the equations $α(x) = k$ for $α\in {\mathcal A}$ and integers $k \in [-m, m]$ is denoted by ${\mathcal A}^m$. It is proved that the coefficients of the characteristic polynomial of ${\mathcal A}^m$ are quasi-polynomials in $m$ and that they satisfy a simple combinatorial reciprocity law.

math.CO↗

arXiv2006

h-vectors of generalized associahedra and non-crossing partitions

A case-free proof is given that the entries of the $h$-vector of the cluster complex $Δ(Φ)$, associated by S. Fomin and A. Zelevinsky to a finite root system $Φ$, count elements of the lattice $\nc$ of noncrossing partitions of corresponding type by rank. Similar interpretations for the $h$-vector of the positive part of $Δ(Φ)$ are provided. The proof utilizes the appearance of the complex $Δ(Φ)$ in the context of the lattice $\nc$, in recent work of two of the authors, as well as an explicit shelling of $Δ(Φ)$.

math.CO↗

arXiv2005

On some enumerative aspects of generalized associahedra

We prove a conjecture of F. Chapoton relating certain enumerative invariants of (a) the cluster complex associated by S. Fomin and A. Zelevinsky to a finite root system and (b) the lattice of noncrossing partitions associated to the corresponding finite real reflection group.

math.CO↗

arXiv2005

Shellability of noncrossing partition lattices

We give a case-free proof that the lattice of noncrossing partitions associated to any finite real reflection group is EL-shellable. Shellability of these lattices was open for the groups of type $D_n$ and those of exceptional type and rank at least three.

math.CO↗

arXiv2003

Ehrhart polynomials, simplicial polytopes, magic squares and a conjecture of Stanley

It is proved that a certain symmetric sequence of nonnegative integers arising in the enumeration of magic squares of given size n by row sums or, equivalently, in the generating function of the Ehrhart polynomial of the polytope of doubly stochastic n by n matrices, is equal to the h-vector of a simplicial polytope and hence that it satisfies the conditions of the g-theorem. The unimodality of this sequance, which follows, was conjectured by Stanley (1983). Several generalizations are given.

math.CO↗

arXiv1997

Extended Linial Hyperplane Arrangements for Root Systems and a Conjecture of Postnikov and Stanley

A hyperplane arrangement is said to satisfy the ``Riemann hypothesis'' if all roots of its characteristic polynomial have the same real part. This property was conjectured by Postnikov and Stanley for certain families of arrangements which are defined for any irreducible root system and was proved for the root system $A_{n-1}$. The proof is based on an explicit formula for the characteristic polynomial, which is of independent combinatorial significance. Here our previous derivation of this formula is simplified and extended to similar formulae for all but the exceptional root systems. The conjecture follows in these cases.

math.CO↗

arXiv1997

A Simple Bijection for the Regions of the Shi Arrangement of Hyperplanes

The Shi arrangement ${\mathcal S}_n$ is the arrangement of affine hyperplanes in ${\mathbb R}^n$ of the form $x_i - x_j = 0$ or $1$, for $1 \leq i < j \leq n$. It dissects ${\mathbb R}^n$ into $(n+1)^{n-1}$ regions, as was first proved by Shi. We give a simple bijective proof of this result. Our bijection generalizes easily to any subarrangement of ${\mathcal S}_n$ containing the hyperplanes $x_i - x_j = 0$ and to the extended Shi arrangements.

math.CO↗

arXiv1997

Piles of Cubes, Monotone Path Polytopes and Hyperplane Arrangements

Monotone path polytopes arise as a special case of the construction of fiber polytopes, introduced by Billera and Sturmfels. A simple example is provided by the permutahedron, which is a monotone path polytope of the standard unit cube. The permutahedron is the zonotope polar to the braid arrangement. We show how the zonotopes polar to the cones of certain deformations of the braid arrangement can be realized as monotone path polytopes. The construction is an extension of that of the permutahedron and yields interesting connections between enumerative combinatorics of hyperplane arrangements and geometry of monotone path polytopes.

math.CO↗