Posets from admissible Coxeter sequences

Abstract

We study the equivalence relation on the set of acyclic orientations of an undi- rected graph Γ 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 asso- ciate 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 Γ to those over Γ′ and Γ′′, the graphs obtained from Γ 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 conju- gacy 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.