Frobenius manifolds on orbits spaces

TL;DR

This paper explores Frobenius manifold structures on orbit spaces of finite group representations, revealing new rational and trivial structures beyond classical invariant theory.

Contribution

It applies Dubrovin's method to various orbit spaces, discovering multiple Frobenius structures, including some previously unknown or unrelated to invariant theory.

Findings

Some orbit spaces admit multiple Frobenius structures.

Identified rational and trivial Frobenius structures not linked to invariant theory.

Extended understanding of Frobenius manifolds beyond Coxeter groups.

Abstract

The orbits space of an irreducible linear representation of a finite group is a variety whose coordinate ring is the ring of invariant polynomials. Boris Dubrovin proved that the orbits space of the standard reflection representation of an irreducible finite Coxeter group $\mathcal W$ acquires a natural polynomial Frobenius manifold structure. We apply Dubrovin's method on various orbits spaces of linear representations of finite groups. We find some of them has non or several natural Frobenius manifold structures. On the other hand, these Frobenius manifold structures include rational and trivial structures which are not known to be related to the invariant theory of finite groups.