Automata, reduced words, and Garside shadows in Coxeter groups

TL;DR

This paper introduces a class of finite automata recognizing reduced words in Coxeter groups, linking automaton minimality to properties of Garside shadows and small roots, advancing understanding of Coxeter group automata.

Contribution

It defines automata based on Garside shadows and weak order, and explores their relation to canonical automata, providing partial proofs for conjectures on automaton minimality.

Findings

Automata associated with the smallest Garside shadow are minimal.

Canonical automaton minimality is characterized by the finiteness of the root system.

Partial positive answers to conjectures on automaton minimality in Coxeter groups.

Abstract

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.