On the symmetry of Stokes regions of a simple singularity
Let \(f(x)\) be a quasihomogeneous polynomial with a simple singularity at \(x=0 \in \mathbb{C}^ n\) of the Milnor number \(k\). Let \(\varphi (x,t)\) with \(t \in T=\mathbb{C}^{k-1}\) be a universal unfolding of \(f(x)\) which we can also assume to be quasihomogeneous. Then for each parameter \(t \in T\) \(\varphi_ t(x)=\varphi (x,t)\) has \(k\) critical values (accounting multiplicities) which we denote by \(w_ 1(t),\dots,w_ k(t)\). We define the following two sets: \[ \begin{aligned} P & =\{t \in T | \exists i \neq j,\text{ s.t. Re} w_ i(t)=\text{Re} w_ j(t)\} \\ Q & =\{t \in T | \exists i \neq j,\text{ s.t. } w_ i(t)=w_ j(t), \text{ or } \exists i,j,l, \text{ s.t. Re} w_ i(t)=\text{Re} w_ j(t)=\text{Re} w_ l(t)\}. \end{aligned} \] The components of \(T \backslash P\) are called Stokes regions and the components of \(P \backslash Q\) are called walls of the Stokes regions which form the smooth part of the boundary of a Stokes region. Two Stokes regions are adjacent if they have a wall in common. We can construct a graph \(\Gamma\) which describes the adjacency of the Stokes regions as follows: take for each Stokes region one vertex and combine two vertices with the same number of edges as the walls that the corresponding Stokes regions have in common. The aim of this paper is to study the symmetry of this graph and to prove the following result: Theorem: If \(f(x)\) has a simple singularity of the type \(A_ k\), then the symmetry group of the graph \(\Gamma\) is the dihedral group \(D_ q\), where \(q=3\) if \(k=2\) and \(q=2(k+1)\) if \(k \geq 3\).
scientific article; zbMATH DE number 537263