Combinatorial structure of Stokes regions of a simple singularity (Q1923235)

Stokes regions naturally arise in many mathematical problems. For a simple singularity, they are defined in the parameter space of a universal unfolding as the connected components of the set of parameters for which the corresponding critical values have pairwise different real parts. This paper studies the combinatorial structure of Stokes regions of a simple singularity, that is, how they are situated in space relative to each other. This structure can be described by constructing a configuration graph as follows: take one vertex for each Stokes region and combine two vertices with an edge if the two corresponding Stokes regions are adjacent. The main result of this paper explains how to construct this graph explicitly by establishing between the set of Stokes regions and the set of certain tuples of reflections generating a Weyl group a bijection which commutes with the braid group actions on the two sets. The arguments depend heavily on the Looijenga-Lyashko covering from the parameter space of a universal unfolding of a simple singularity to a space of polynomials and use a result of P. Deligne about the braid group action on the above set of reflection tuples.