In 1993 Brink and Howlett proved that finitely generated Coxeter groups are automatic. In particular, they constructed a finite state automaton recognizing the language of reduced words in the Coxeter group. This automaton is constructed in terms of the remarkable set of "elementary roots" in the associated root system.
In this talk we outline the construction of Brink and Howlett. We also describe the minimal automaton recognizing the language of reduced words, and prove necessary and sufficient conditions for the Brink-Howlett automaton to coincide with this minimal automaton. This resolves a conjecture of Hohlweg, Nadeau, and Williams, and is joint work with Yeeka Yau.
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