Automata, reduced words and Garside shadows in Coxeter groups

作者：Hohlweg Christophe; Nadeau Philippe; Williams Nathan

来源：Journal of Algebra, 2016, 457: 431-456.

DOI：10.1016/j.jalgebra.2016.04.006

• Coxeter groups
• Garside shadows
• Low elements
• Weak order
• Dominance order
• Small roots
• Automaton and reduced words

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.

• 出版日期2016-7-1