A poset connected to Artin monoids of simply laced type

@article{Cohen2006APC,
  title={A poset connected to Artin monoids of simply laced type},
  author={Arjeh M. Cohen and Di{\'e} A. H. Gijsbers and David B. Wales},
  journal={J. Comb. Theory, Ser. A},
  year={2006},
  volume={113},
  pages={1646-1666}
}
Let W be a Weyl group whose type is a simply laced Dynkin diagram. On several W-orbits of sets of mutually commuting reflections, a poset is described which plays a role in linear representations of the corresponding Artin group A. The poset generalizes many properties of the usual order on positive roots of W given by height. In this paper, a linear representation of the positive monoid of A is defined by use of the poset.