Posets from Admissible Coxeter Sequences

Details Posets from Admissible Coxeter Sequences Posets from Admissible Coxeter Sequences

2009-10-22

arxiv.org/abs/0910.4376v2

Electron. J. Combin. 18(1) (2011) #R197

Mathematics

Combinatorics

16 pages, 4 figures. Several examples have been added

Scientific paper

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.

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