Admissible submonoids of Artin-Tits monoids

TL;DR

This paper extends Muhlherr's results to Artin-Tits monoids and groups of spherical type, classifying admissible partitions and LCM-homomorphisms, unifying various substructure concepts within these algebraic objects.

Contribution

It generalizes Muhlherr's Coxeter group results to Artin-Tits monoids and groups, providing a unified classification of admissible partitions and LCM-homomorphisms.

Findings

Classification of admissible partitions with finite labels

Extension of submonoid and subgroup structures

Complete classification of LCM-homomorphisms

Abstract

We show the analogue of Muhlherr's [Coxeter groups in Coxeter groups, Finite Geom. and Combinatorics, Cambridge Univ. Press (1993), 277-287] for Artin-Tits monoids, and for Artin-Tits groups of spherical type. That is, the submonoid (resp. subgroup) of an Artin-Tits monoid (resp. group of spherical type) induced by an admissible partition of the Coxeter graph is an Artin-Tits monoid (resp. group). This generalizes and unifies the situation of the submonoid (resp. subgroup) of fixed elements of an Artin-Tits monoid (resp. group of spherical type) under the action of graph automorphisms, and the notion of LCM-homomorphisms defined by Crisp in [Injective maps between Artin groups, Geom. Group Theory Down Under, Canberra (1996) 119-137] and generalized by Godelle in [Morphismes injectifs entre groupes d'Artin-Tits, Algebr. Geom. Topol. 2 (2002), 519--536]. We then complete the classification of the admissible partitions for which the Coxeter graphs involved have no infinite label, started by Muhlherr in [Some contributions to the theory of buildings based on the gate property, Dissertation, T\"ubingen (1994)]. This leads us to the classification of Crisp's LCM-homomorphisms.