Words with intervening neighbours in infinite Coxeter groups are reduced
Henrik Eriksson, Kimmo Eriksson

TL;DR
This paper proves that words with the intervening neighbours property in infinite Coxeter groups are irreducible, providing a shorter proof using the root automaton for recognition.
Contribution
It offers a new, shorter proof of irreducibility of certain words in infinite Coxeter groups using the root automaton method.
Findings
Words with the intervening neighbours property are irreducible in infinite Coxeter groups.
A new proof technique using the root automaton is introduced.
The proof simplifies previous arguments for irreducibility.
Abstract
Consider a graph with vertex set S. A word in the alphabet S has the intervening neighbours property if any two occurrences of the same letter are separated by all its graph neighbours. For a Coxeter graph, words represent group elements. Speyer recently proved that words with the intervening neighbours property are irreducible if the group is infinite and irreducible. We present a new and shorter proof using the root automaton for recognition of irreducible words.
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Taxonomy
Topicssemigroups and automata theory · DNA and Biological Computing · Advanced Combinatorial Mathematics
