Flat coordinates of algebraic Frobenius manifolds in small dimensions

Abstract

Orbit spaces of the reflection representation of finite irreducible Coxeter groups provide polynomial Frobenius manifolds. Flat coordinates of the Frobenius metric $\eta$ are Saito polynomials which are distinguished basic invariants of the Coxeter group. Algebraic Frobenius manifolds are typically related to quasi-Coxeter conjugacy classes in finite Coxeter groups. We find explicit relations between flat coordinates of the Frobenius metric $\eta$ and flat coordinates of the intersection form $g$ for most known examples of algebraic Frobenius manifolds up to dimension 4. In all the cases, flat coordinates of the metric $\eta$ appear to be algebraic functions on the orbit space of the Coxeter group.