Flat coordinates of algebraic Frobenius manifolds in small dimensions
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.
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