Morphisms and order ideals of toric posets

Toric posets are cyclic analogues of finite posets. They can be viewed combinatorially as equivalence classes of acyclic orientations generated by converting sources into sinks, or geometrically as chambers of toric graphic hyperplane arrangements. In this paper we study toric intervals, morphisms, and order ideals, and we provide a connection to cyclic reducibility and conjugacy in Coxeter groups.Comment: 28 pages, 8 figures. A 12-page "extended abstract" version appears as [v2

A geometric approach to acyclic orientations

The set of acyclic orientations of a connected graph with a given sink has a natural poset structure. We give a geometric proof of a result of Jim Propp: this poset is the disjoint union of distributive lattices. Let G be a connected graph on the vertex set [n] = {0} ∪ [n], where [n] denotes the set {1,...,n}. Let P denote the collection of acyclic orientations of G, and let P0 denote the collection of acyclic orientations of G with 0 as a sink. If Ω is an orientation in P with the vertex i as a source, we can obtain a new orientation Ω ′ with i as a sink by firing the vertex i, reorienting all the edges adjacent to i towards i. The orientations Ω and Ω ′ agree away from i. A firing sequence from Ω to Ω ′ in P consists of a sequence Ω = Ω1,...,Ωm+1 = Ω ′ of orientations and a function F: [m] − → [n] such that for each i ∈ [m], the orientation Ωi+1 is obtained from Ωi by firing the vertex F(i). We will abuse language by calling F itself a firing sequence. We make P into a preorder by writing Ω ≤ Ω ′ if and only if there is a firing sequence from Ω to Ω ′. From the definition it is clear that P is reflexive and transitive. While P is only a preorder, P0 is a poset. By finiteness, antisymmetry can be verified by showing that firing sequences in P0 cannot be arbitrarily long. Thi