The Construction Problem of Algebraic Potentials and Reflection Groups

This paper has two aims. The first one is the construction problem of algebraic potentials of Frobenius manifolds. We show examples of such potentials for the cases of reflection groups of types $H_4,E_6,E_7,E_8$ and also include those which are already known. The second one is an application of such potentials to singularity theory. We introduce families of hypersurfaces of ${\bf C}^3$ which are deformations of $E_n$-singularities $(n=6,7,8)$ but are not the versal families of $E_n$-singularities. We study the properties of the families. In particular we show the correspondence between such families and the algebraic potentials constructed in the first aim. Moreover we discuss the relationship between the complex reflection groups $ST33$ and $ST34$ and the two families corresponding to the $E_6$-singularity and the $E_7$-singularity.