Question: Is is possible to realise the positive root lattice $\Phi_{\Delta}^{>0}$ (viewed as an abstract poset) of a root system $\Phi_\Delta$ associated to a Dynkin or affine Dynkin diagram $\Delta$ in terms of the set of reflections of the corresponding Weyl group $W_\Delta$ together with a choice of coxeter element ? In other words is it possible to define a partial order $\leq_{\, c}$ on the set of all reflections in $W_\Delta$  depending only upon on a choice of coxeter element $c$  which is isomorphic to $\Phi_\Delta^{>0}$ as a poset.
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$\begingroup$ I should qualify my question by requesting that the construction of $\leq_{\, c}$ should be systematic enough so that it can be implemented for a general coxeter group. best, A. Leverkühn. $\endgroup$ – A. Leverkuhn Apr 18 '16 at 1:03

2$\begingroup$ Concerning the lattice: do you mean the poset of positive roots with $\alpha \leq \beta$ if $\beta  \alpha \in \mathbb{Z}_{\geq 0} \Delta$ with simple system $\Delta$? This poset is not a lattice. Or do you mean its distributive lattice given by all order ideals ordered by containment? $\endgroup$ – Christian Stump Apr 18 '16 at 7:55

2$\begingroup$ Why do you then call it "positive root lattice"? $\endgroup$ – Christian Stump Apr 18 '16 at 15:24

2$\begingroup$ There is a different paper of Christian's which is relevant to the motivation for the question: arxiv.org/abs/1212.2876. According to this paper, there is no reasonable candidate for a "poset of positive roots" in $H_4$ (though note that there are reasonable candidates for dihedral groups and for $H_3$, due to Drew Armstrong). Thus, the hope that one could carry out the procedure for noncrystallographic Coxeter groups seems doomed. (I still think it's an interesting question on its own merits, though.) $\endgroup$ – Hugh Thomas Apr 18 '16 at 16:58

1$\begingroup$ The subject of the paper by Christian (and Michael Cuntz) that I referred to in my previous comment is exactly the question of whether it is possible to find a poset associated to $H_4$ "whose ideal structure returns the notion and basic properties of nonnesting partitions". $\endgroup$ – Hugh Thomas Apr 18 '16 at 18:12