Frobenius manifolds associated to Coxeter groups of type E_7 and E_8

TL;DR

This paper computes flat coordinates and Frobenius potentials for manifolds associated with the exceptional Coxeter groups E_7 and E_8, expanding the understanding of their algebraic and geometric structures.

Contribution

It provides explicit calculations of flat coordinates and Frobenius potentials for E_7 and E_8, which were previously not fully determined.

Findings

Explicit flat coordinates for E_7 and E_8 Frobenius manifolds

Derivation of Frobenius potentials for these groups

Enhanced understanding of algebraic structures of exceptional Coxeter groups

Abstract

Flat coordinates for Frobenius manifolds defined on the orbit space of a Coxeter group W are specified through a certain system of generators of W-invariant polynomials. In this note, starting from basic invariants proposed by M.Mehta, we calculate flat coordinates for the exceptional groups of type E_7 and E_8, leading to a derivation of the potentials for the associated Frobenius structures.