# the root lattice, reflections, and a coxeter element

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.

• 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. Apr 18, 2016 at 1:03
• 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? Apr 18, 2016 at 7:55
• Why do you then call it "positive root lattice"? Apr 18, 2016 at 15:24
• 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 non-crystallographic Coxeter groups seems doomed. (I still think it's an interesting question on its own merits, though.) Apr 18, 2016 at 16:58
• 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 non-nesting partitions". Apr 18, 2016 at 18:12