Additive relative invariants and the components of a linear free divisor

TL;DR

This paper explores the structure of linear free divisors in prehomogeneous vector spaces, showing that additive functions are trivial in certain cases and relating the number of divisor components to group properties.

Contribution

It proves that for linear free divisors, the associated abelian quotient is an algebraic torus and relates divisor components to the group's abelianization, extending existing results.

Findings

Additive functions of H are trivial for linear free divisors.

Number of divisor components equals the dimension of the group's abelianization.

Simplified proofs for cases where G is abelian, reductive, or solvable.

Abstract

A 'prehomogeneous vector space' is a rational representation $\rho:G\to\mathrm{GL}(V)$ of a connected complex linear algebraic group $G$ that has a Zariski open orbit $\Omega\subset V$. Mikio Sato showed that the hypersurface components of $D:=V\setminus \Omega$ are related to the rational characters $H\to\mathrm{GL}(\mathbb{C})$ of $H$, an algebraic abelian quotient of $G$. Mimicking this work, we investigate the 'additive functions' of $H$, the homomorphisms $\Phi:H\to (\mathbb{C},+)$. Each such $\Phi$ is related to an 'additive relative invariant', a rational function $h$ on $V$ such that $h\circ \rho(g)-h=\Phi(g)$ on $\Omega$ for all $g\in G$. Such an $h$ is homogeneous of degree $0$, and helps describe the behavior of certain subsets of $D$ under the $G$--action. For those prehomogeneous vector spaces with $D$ a type of hypersurface called a linear free divisor, we prove there are no nontrivial additive functions of $H$, and hence $H$ is an algebraic torus. From this we gain insight into the structure of such representations and prove that the number of irreducible components of $D$ equals the dimension of the abelianization of $G$. For some special cases ($G$ abelian, reductive, or solvable, or $D$ irreducible) we simplify proofs of existing results. We also examine the homotopy groups of $V\setminus D$.