Dominant regions in noncrystallographic hyperplane arrangements

For a crystallographic root system, dominant regions in the Catalan hyperplane arrangement are in bijection with antichains in a partial order on the positive roots. For a noncrystallographic root system, the analogous arrangement and regions have importance in the representation theory of an associated graded Hecke algebra. Since there is also an analogous root order, it is natural to hope that a similar bijection can be used to understand these regions. We show that such a bijection does hold for type H₃ and for type I₂(m), including arbitrary ratio of root lengths when m is even, but does not hold for type H₄. We give a criterion that explains this failure and a list of the 16 antichains in the H₄ root order which correspond to empty regions.