Abstract
Toric posets are in some sense a natural “cyclic” version of finite posets in that they capture the fundamental features of a partial order but without the notion of minimal or maximal elements. They can be thought of combinatorially as equivalence classes of acyclic orientations under the equivalence relation generated by converting sources into sinks, or geometrically as chambers of toric graphic hyperplane arrangements. In this paper, we define toric intervals and toric order-preserving maps, which lead to toric analogues of poset morphisms and order ideals. We develop this theory, discuss some fundamental differences between the toric and ordinary cases, and outline some areas for future research. Additionally, we provide a connection to cyclic reducibility and conjugacy in Coxeter groups.
Highlights
A finite poset can be described by at least one directed acyclic graph where the elements are vertices and directed edges encode relations
Equivalence classes are called toric posets. These objects have arisen in a variety of contexts in the literature, including but not limited to chip-firing games [1], Coxeter groups [2,3,4], graph polynomials [5], lattices [6,7], and quiver representations [8]. These equivalence classes were first formalized as toric posets in [9], where the effort was made to develop a theory of these objects in conjunction with the existing theory of ordinary posets
A( G ), a toric poset corresponds to a chamber of a toric graphic arrangement Ator ( G ) = q(A( G )), which q is the image of A( G ) under the quotient map RV −→ RV /ZV
Summary
A finite poset can be described by at least one directed acyclic graph where the elements are vertices and directed edges encode relations. For most features of toric posets, i.e., the toric analogues of standard posets features, the geometric viewpoint is needed to see the natural proper definitions and to prove structure theorems. Once this is done, the definitions and characterizations frequently have simple combinatorial (non-geometric) interpretations. Toric posets have no such binary relation, and so this is why we need to go to the geometric setting to define the basic features. This analogy is not perfect, because toric posets are not a generalization of ordinary posets like how topological spaces extend metric spaces It should motivate the reliance on geometric methods throughout this paper.
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