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A prehomogeneous vector space is a rational representation $\rho:G\to GL(V)$
of a connected complex linear algebraic group $G$ that has a Zariski open orbit
$\Omega\subset V$. M. Sato showed that the hypersurface components of
$D:=V\setminus \Omega$ are related to the characters $H\to GL(\mathbb{C})$ of
$H$, an algebraic abelian quotient of $G$. Mimicking this work, we investigate
the additive functions of $H$, 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 describes 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
$V\setminus D$.

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you can request a copy directly from the author.

ResearchGate has not been able to resolve any citations for this publication.

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.

A complex hypersurface D in complex affine n-space C^n is a linear free divisor (LFD) if its module of logarithmic vector fields has a global basis of linear vector fields. We classify all LFDs for n at most 4. Analogous to Grothendieck's comparison theorem, we say that the global logarithmic comparison theorem (GLCT) holds for D if the complex of global logarithmic differential forms computes the complex cohomology of the complement of D in C^n. We develop a general criterion for the GLCT for LFDs and prove that it is fulfilled whenever the Lie algebra of linear logarithmic vector fields is reductive. For n at most 4, we show that the GLCT holds for all LFDs. We show that LFDs arising naturally as discriminants in quiver representation spaces (of real Schur roots) fulfill the GLCT. As a by-product we obtain a simplified proof of a theorem of V. Kac on the number of irreducible components of such discriminants. Comment: 46 pages, 2 figures, 5 tables, final version

Linear free divisors are free divisors, in the sense of K.Saito, with linear presentation matrix (example: normal crossing divisors). Using techniques of deformation theory on representations of quivers, we exhibit families of linear free divisors as discriminants in representation spaces for real Schur roots of a finite quiver. We review some basic material on quiver representations, and explain in detail how to verify whether the discriminant is a free divisor and how to determine its components and their equations, using techniques of A. Schofield. As an illustration, the linear free divisors that arise as the discriminant from the highest roots of Dynkin quivers of type E7 and E8 are treated explicitly.

We prove that if D D is a “strongly quasihomogeneous" free divisor in the Stein manifold X X , and U U is its complement, then the de Rham cohomology of U U can be computed as the cohomology of the complex of meromorphic differential forms on X X with logarithmic poles along D D , with exterior derivative. The class of strongly quasihomogeneous free divisors, introduced here, includes free hyperplane arrangements and the discriminants of stable mappings in Mather’s nice dimensions (and in particular the discriminants of Coxeter groups).

We apply previous results on the representations of solvable linear algebraic groups to construct a new class of free divisors whose complements are K(π,1)ʼs. These free divisors arise as the exceptional orbit varieties for a special class of “block representations” and have the structure of determinantal arrangements.Among these are the free divisors defined by conditions for the (modified) Cholesky-type factorizations of matrices, which contain the determinantal varieties of singular matrices of various types as components. These complements are proven to be homotopy tori, as are the Milnor fibers of these free divisors. The generators for the complex cohomology of each are given in terms of forms defined using the basic relative invariants of the group representation.

Some remarks on linear free divisors, E-mail to Ragnar-Olaf Buchweitz

- Michel Brion

Michel Brion, Some remarks on linear free divisors, E-mail to Ragnar-Olaf Buchweitz, September 2006. 13

Solvable groups, free divisors and nonisolated matrix singularities I: Towers of free divisors

- James Damon
- Brian Pike

James Damon and Brian Pike, Solvable groups, free divisors and nonisolated matrix
singularities I: Towers of free divisors, Submitted. arXiv:1201.1577 [math.AG], 2012.
2, 12, 18, 22

Translated from the 1998 Japanese original by Makoto Nagura and Tsuyoshi Niitani and revised by the author

- Tatsuo Kimura

Tatsuo Kimura, Introduction to prehomogeneous vector spaces, Translations of
Mathematical Monographs, vol. 215, American Mathematical Society, Providence,
RI, 2003, Translated from the 1998 Japanese original by Makoto Nagura and
Tsuyoshi Niitani and revised by the author. MR 1944442 (2003k:11180) 1, 3, 4,
6, 20

- Brian Pike

Brian Pike, On Fitting ideals of logarithmic vector fields and Saito's criterion,
arXiv:1309.3769 [math.AG]. 9, 13