Posets from Admissible Coxeter Sequences

TL;DR

This paper explores the structure of acyclic orientations of graphs through posets derived from Coxeter theory, providing new combinatorial invariants and solutions to the conjugacy problem in simply-laced Coxeter groups.

Contribution

It introduces a novel poset construction from admissible Coxeter sequences and establishes a bijection that simplifies understanding equivalence classes and solves the conjugacy problem.

Findings

Posets characterize equivalence classes of acyclic orientations.

Bijection relates graphs via edge deletion and contraction.

Provides an elegant proof of the conjugacy problem solution.

Abstract

We study the equivalence relation on the set of acyclic orientations of an undirected graph G generated by source-to-sink conversions. These conversions arise in the contexts of admissible sequences in Coxeter theory, quiver representations, and asynchronous graph dynamical systems. To each equivalence class we associate a poset, characterize combinatorial properties of these posets, and in turn, the admissible sequences. This allows us to construct an explicit bijection from the equivalence classes over G to those over G' and G", the graphs obtained from G by edge deletion and edge contraction of a fixed cycle-edge, respectively. This bijection yields quick and elegant proofs of two non-trivial results: (i) A complete combinatorial invariant of the equivalence classes, and (ii) a solution to the conjugacy problem of Coxeter elements for simply-laced Coxeter groups. The latter was recently proven by H. Eriksson and K. Eriksson using a much different approach.