A Poset Connected to Artin Monoids of Simply Laced Type
  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 representatons 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. 
A. M. Cohen, D. A. H. Gijsbers, D. B. Wales