Discriminants and Jacobians of virtual reflection groups

TL;DR

This paper explores the factorization of Jacobians and discriminants in polynomial extensions, generalizing classical invariant theory of reflection groups to a broader class called virtual reflection groups.

Contribution

It introduces the concept of well-ramified extensions and their discriminants, extending invariant theory beyond classical reflection groups to virtual reflection groups.

Findings

Factorization of Jacobian into irreducibles in polynomial extensions

Definition and analysis of well-ramified extensions and their discriminants

Framework parallels classical invariant theory for complex reflection groups

Abstract

Let A be a polynomial algebra with complex coefficients. Let B be a finite extension ring of A which is also a polynomial algebra. We describe the factorisation of the Jacobian J of the extension into irreducibles. We also introduce the notion of a well-ramified extension and define its discriminant polynomial D. In the particular case where A is the ring of invariants of B under the action of a group (i.e., a Galois extension), this framework corresponds to the classical invariant theory of complex reflection groups. In the more general case of a well-ramified extension, we explain how the pair (D,J) behaves similarly to a Galois extension. This work can be viewed as the first step towards a possible invariant theory of "virtual reflection groups".