Elementary roots and admissible subsets of Coxeter groups

Abstract

Admissibility of a subset S′ of a Coxeter system (W, S) is a condition implying that S′ is the set of Coxeter generators of a Coxeter subgroup W′ of W, in such a way that the root system of W′, as permutation set, abstractly embeds in that of W. We give an algorithm determining whether a subset S′ is admissible, in terms of a (previously known) finite state automaton which is constructed using the set of elementary roots of Brink and Howlett.