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{\bf Matthew Macauley and Henning S. Mortveit }
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{\bf Posets from Admissible Coxeter Sequences}
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We study the equivalence relation on the set of acyclic orientations
of an undirected graph $\Gamma$ 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
$\Gamma$ to those over $\Gamma'$ and $\Gamma''$, the graphs obtained
from $\Gamma$ 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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