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.