Free divisors in prehomogeneous vector spaces

TL;DR

This paper investigates linear free divisors arising from prehomogeneous vector spaces, especially in quiver representations, providing characterizations, properties, and classifications of such divisors.

Contribution

It characterizes prehomogeneous vector spaces with linear free divisors, classifies reductive cases, and links quiver structures to the existence of these divisors.

Findings

Reductive linear free divisors have matching geometric and representation-theoretic components.

Quivers with cycles do not produce linear free divisors.

All tame quiver linear free divisors are locally weakly quasihomogeneous.

Abstract

We study linear free divisors, that is, free divisors arising as discriminants in prehomogeneous vector spaces, and in particular in quiver representation spaces. We give a characterization of the prehomogeneous vector spaces containing such linear free divisors. For reductive linear free divisors, we prove that the numbers of geometric and representation theoretic irreducible components coincide. As a consequence, we find that a quiver can only give rise to a linear free divisor if it has no (oriented or unoriented) cycles. We also deduce that the linear free divisors which appear in Sato and Kimura's list of irreducible prehomogeneous vector spaces are the only irreducible reductive linear free divisors. Furthermore, we show that all quiver linear free divisors are strongly Euler homogeneous, that they are locally weakly quasihomogeneous at points whose corresponding representation is not regular, and that all tame quiver linear free divisors are locally weakly quasihomogeneous. In particular, the latter satisfy the logarithmic comparison theorem.