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Automata, reduced words, and Garside shadows in Coxeter groups
In this article, we introduce and investigate a class of finite deterministic
automata that all recognize the language of reduced words of a finitely
generated Coxeter system (W,S). The definition of these automata is
straightforward as it only requires the notion of weak order on (W,S) and the
related notion of Garside shadows in (W,S), an analog of the notion of a
Garside family. Then we discuss the relations between this class of automata
and the canonical automaton built from Brink and Howlett's small roots. We end
this article by providing partial positive answers to two conjectures: (1) the
automata associated to the smallest Garside shadow is minimal; (2) the
canonical automaton is minimal if and only if the support of all small roots is
spherical, i.e., the corresponding root system is finite.Comment: 21 pages, 7 figures; v2: 23 pages, 8 figures, Remark 3.15 added,
accepted in Journal of Algebra, computational sectio
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