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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.

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... In §5, we study prehomogeneous vector spaces with the property that D := V \Ω is a type of hypersurface called a linear free divisor. Such objects have been of much interest recently (e.g., [BM06,GMNRS09,GMS11,DP12a,DP12b]), and were our original motivation. Using a criterion due to Brion and the results of §3.6, we show in Theorem 5.6 that such prehomogeneous vector spaces have no nontrivial additive relative invariants or nontrivial additive functions. ...

... Thus, the homotopy groups may largely be computed from the homotopy groups of the semisimple part [L, L] of G. For instance, if G is solvable then [L, L] = {e} and hence by Proposition 5.29, the spaces Ω, P , and K are K(π, 1)'s, as shown in [DP12a]. ...

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
$V\setminus D$.

... Buchweitz and Mond [3] showed that the arrangement defined by the product of the maximal minors of a n × (n + 1) matrix of indeterminates is free. Recently, Damon and Pike [4] show that certain determinantal arrangements coming from symmetric, skew-symmetric and square general matrices are free and have complements that are K(π, 1). In both of these cases, the arrangements turn out to be linear free divisors (i.e. the basis for Der X (− log D) is generated by linear vector fields). ...

We explore a natural extension of braid arrangements in the context of
determinantal arrangements. We show that these determinantal arrangements are
free divisors. Additionally, we prove that free determinantal arrangements
defined by the minors of $2\times n$ matrices satisfy nice combinatorial
properties.
We also study the topology of the complements of these determinantal
arrangements, and prove that their higher homotopy groups are isomorphic to
those of $S^3$. Furthermore, we find that the complements of arrangements
satisfying those same combinatorial properties above have Poincar\'e
polynomials that factor nicely.

... In the Saito-holonomic case, the local product structure implies that the logarithmic stratification refines the canonical Whitney stratification and is Whitney itself, compare [14,Prop. 3.11] and [15] for further details. ...

For a germ $f$ on a complex manifold $X$, we introduce a complex derived from
the Liouville form acting on logarithmic differential forms, and give an
exactness criterion. We use this logarithmic complex to connect properties of
the $D$-module generated by $f^s$ to homological data of the Jacobian ideal;
specifically in many cases we show that the annihilator of $f^s$ is generated
by derivations. Moreover, through local cohomology, we connect the cohomology
of the Milnor fiber to the Jacobian module through logarithmic differentials
under blow-ups.
In particular, we consider (not necessarily reduced) hyperplane arrangements:
we prove many cases of the conjecture of Terao on the annihilator of $1/f$; we
disprove a corresponding conjecture on the annihilator of $f^s$; we show that
the Bernstein--Sato polynomial of an arrangement is not determined by its
intersection lattice; we prove that arrangements that satisfy Terao's
conjecture fulfill the Strong Monodromy Conjecture, and that this includes as
very special cases all arrangements of Coxeter and of crystallographic type,
and all multi-arrangements in dimension 3.

For a germ of a variety $\mathcal{V}, 0 \subset \mathbb C^N, 0$, a singularity $\mathcal{V}_0$ of ‘type $\mathcal{V}$’ is given by a germ $f_0 : \mathbb C^n, 0 \to \mathbb C^N, 0$ which is transverse to $\mathcal{V}$ in an appropriate sense so that $\mathcal{V}_0 = f_0^{\,-1}(\mathcal{V})$. If $\mathcal{V}$ is a hypersurface germ, then so is $\mathcal{V}_0 $, and by transversality ${\operatorname{codim}}_{\mathbb C} {\operatorname{sing}}(\mathcal{V}_0) = {\operatorname{codim}}_{\mathbb C} {\operatorname{sing}}(\mathcal{V})$ provided $n > {\operatorname{codim}}_{\mathbb C} {\operatorname{sing}}(\mathcal{V})$. So $\mathcal{V}_0, 0$ will exhibit singularities of $\mathcal{V}$ up to codimension n.
For singularities $\mathcal{V}_0, 0$ of type $\mathcal{V}$, we introduce a method to capture the contribution of the topology of $\mathcal{V}$ to that of $\mathcal{V}_0$. It is via the ‘characteristic cohomology’ of the Milnor fiber (for $\mathcal{V}, 0$ a hypersurface), and complement and link of $\mathcal{V}_0$ (in the general case). The characteristic cohomology of the Milnor fiber $\mathcal{A}_{\mathcal{V}}(\,f_0; R)$, and respectively of the complement $\mathcal{C}_{\mathcal{V}}(\,f_0; R)$, are subalgebras of the cohomology of the Milnor fibers, respectively the complement, with coefficients R in the corresponding cohomology. For a fixed $\mathcal{V}$, they are functorial over the category of singularities of type $\mathcal{V}$. In addition, for the link of $\mathcal{V}_0$ there is a characteristic cohomology subgroup $\mathcal{B}_{\mathcal{V}}(\,f_0, \mathbf{k})$ of the cohomology of the link over a field $\mathbf{k}$ of characteristic 0. The cohomologies $\mathcal{C}_{\mathcal{V}}(\,f_0; R)$ and $\mathcal{B}_{\mathcal{V}}(\,f_0, \mathbf{k})$ are shown to be invariant under the $\mathcal{K}_{\mathcal{V}}$-equivalence of defining germs f0, and likewise $\mathcal{A}_{\mathcal{V}}(\,f_0; R)$ is shown to be invariant under the $\mathcal{K}_{H}$-equivalence of f0 for H the defining equation of $\mathcal{V}, 0$.
We give a geometric criterion involving ‘vanishing compact models’ for both the Milnor fibers and complements which detect non-vanishing subalgebras of the characteristic cohomologies, and subgroups of the characteristic cohomology of the link. Also, we consider how in the hypersurface case the cohomology of the Milnor fiber is a module over the characteristic cohomology $\mathcal{A}_{\mathcal{V}}(\,f_0; R)$. We briefly consider the application of these results to a number of cases of singularities of a given type. In part II, we specialize to the case of matrix singularities and using results on the topology of the Milnor fibers, complements and links of the varieties of singular matrices obtained in another paper allow us to give precise results for the characteristic cohomology of all three types.

For a germ of a variety $\mathcal{V}, 0 \subset \mathbb C^N, 0$, a singularity $\mathcal{V}_0$ of type $\mathcal{V}$, is given by a germ $f_0 : \mathbb C^n, 0 \to \mathbb C^N, 0$ which is transverse to $\mathcal{V}$ in an appropriate sense so that $\mathcal{V}_0 = f_0^{-1}(\mathcal{V})$. For these singularities, we introduce "characteristic cohomology" to capture the contribution of the topology of $\mathcal{V}$ to that of $\mathcal{V}_0$, for the Milnor fiber (for $\mathcal{V}, 0$ a hypersurface), and complement and link of $\mathcal{V}_0$ (in the general case). The characteristic cohomology of both the Milnor fiber and complement are subalgebras of the cohomology of the Milnor fibers, respectively the complement. For a fixed $\mathcal{V}$, they are functorial over the category of singularities of type $\mathcal{V}$. In addition, for the link of $\mathcal{V}_0$ there is a characteristic cohomology subgroup of the cohomology of the link over a field of characteristic 0. The characteristic cohomologies for Milnor fiber and complement are shown to be invariant under the $\mathcal K_{\mathcal{V}}$-equivalence of defining germs $f_0$, resp. for the link invariant under the $\mathcal K_{H}$-equivalence of $f_0$ for $H$ the defining equation of $\mathcal V, 0$. We give a geometric criteria involving "vanishing compact models", which detect nonvanishing subalgebras of the characteristic cohomologies, resp. subgroups for the link. In part II of this paper we specialize to the case of square matrix singularities, which may be general, symmetric or skew-symmetric.

We consider the topology for a class of hypersurfaces with highly nonisolated
singularites which arise as exceptional orbit varieties of a special class of
prehomogeneous vector spaces, which are representations of linear algebraic
groups with open orbits. These hypersurface singularities include both
determinantal hypersurfaces and linear free (and free*) divisors. Although
these hypersurfaces have highly nonisolated singularities, we determine the
topology of their Milnor fibers, complements and links. We do so by using the
action of linear algebraic groups beginning with the complement, instead of
using Morse type arguments on the Milnor fibers. This includes replacing the
local Milnor fiber by a global Milnor fiber which has a complex geometry
resulting from a transitive action of an appropriate algebraic group, yielding
a compact model submanifold for the homotopy type of the Milnor fiber. The
topology includes the (co)homology (in characteristic 0, and 2 torsion in one
family) and homotopy groups, and we deduce the triviality of the monodromy
transformations on rational (or complex) cohomology. The cohomology of the
Milnor fibers and complements are isomorphic as algebras to exterior algebras
or for one family, modules over exterior algebras; and cohomology of the link
is, as a vector space, a truncated and shifted exterior algebra, for which the
cohomology product structure is essentially trivial. We also deduce from Bott's
periodicity theorem, the homotopy groups of the Milnor fibers for determinantal
hypersurfaces in the stable range as the stable homotopy groups of the
associated infinite dimensional symmetric spaces. Applying a Theorem of Oka we
obtain a class of formal linear combinations of exceptional orbit hypersurfaces
which have Milnor fibers which are homotopy equivalent to joins of the compact
model submanifolds.

In this paper we use the results from the first part to compute the vanishing
topology for matrix singularities based on certain spaces of matrices. We place
the variety of singular matrices in a geometric configuration of free divisors
which are the "exceptional orbit varieties" for repesentations of solvable
groups. Because there are towers of representations for towers of solvable
groups, the free divisors actually form a tower of free divisors $E_n$, and we
give an inductive procedure for computing the vanishing topology of the matrix
singularities. The inductive procedure we use is an extension of that
introduced by L\^{e}-Greuel for computing the Milnor number of an ICIS. Instead
of linear subspaces, we use free divisors arising from the geometric
configuration and which correspond to subgroups of the solvable groups.
Here the vanishing topology involves a singular version of the Milnor fiber;
however, it still has the good connectivity properties and is homotopy
equivalent to a bouquet of spheres, whose number is called the singular Milnor
number. We give formulas for this singular Milnor number in terms of singular
Milnor numbers of various free divisors on smooth subspaces, which can be
computed as lengths of determinantal modules. In addition to being applied to
symmetric, general and skew-symmetric matrix singularities, the results are
also applied to compute the Milnor number of isolated Cohen-Macaulay surface
singularities in $\C^4$ and the difference of Betti numbers of Milnor fibers
for isolated Cohen-Macaulay 3-fold singularities in $\C^5$.

This paper is the first part of a two part paper which introduces a method
for determining the vanishing topology of nonisolated matrix singularities. A
foundation for this is the introduction of a method for obtaining new classes
of free divisors from representations $V$ of connected solvable linear
algebraic groups $G$. For equidimensional representations where $\dim G=\dim
V$, with $V$ having an open orbit, we give sufficient conditions that the
complement $E$ of this open orbit, the "exceptional orbit variety", is a free
divisor (or a weaker free* divisor).
We do so by introducing the notion of a "block representation" which is
especially suited for both solvable groups and extensions of reductive groups
by them. This is a representation for which the matrix representing a basis of
associated vector fields on $V$ defined by the representation can be expressed
using a basis for $V$ as a block triangular matrix, with the blocks satisfying
certain nonsingularity conditions. We use the Lie algebra structure of $G$ to
identify the blocks, the singular structure, and a defining equation for $E$.
This construction naturally fits within the framework of towers of Lie groups
and representations yielding a tower of free divisors which allows us to
inductively represent the variety of singular matrices as fitting between two
free divisors. We specifically apply this to spaces of matrices including
$m\times m$ symmetric, skew-symmetric or general matrices, where we prove that
both the classical Cholesky factorization of matrices and a "modified Cholesky
factorization" which we introduce are given by block representations of
solvable group actions. For skew-symmetric matrices, we use an extension of the
method on an infinite dimensional solvable Lie algebra.
In part II, we use these results to compute the vanishing topology for matrix
singularities for all of the classes.

In this paper we use the results from the first part to compute the vanishing
topology for matrix singularities based on certain spaces of matrices. We place
the variety of singular matrices in a geometric configuration of free divisors
which are the "exceptional orbit varieties" for repesentations of solvable
groups. Because there are towers of representations for towers of solvable
groups, the free divisors actually form a tower of free divisors $E_n$, and we
give an inductive procedure for computing the vanishing topology of the matrix
singularities. The inductive procedure we use is an extension of that
introduced by L\^{e}-Greuel for computing the Milnor number of an ICIS. Instead
of linear subspaces, we use free divisors arising from the geometric
configuration and which correspond to subgroups of the solvable groups.
Here the vanishing topology involves a singular version of the Milnor fiber;
however, it still has the good connectivity properties and is homotopy
equivalent to a bouquet of spheres, whose number is called the singular Milnor
number. We give formulas for this singular Milnor number in terms of singular
Milnor numbers of various free divisors on smooth subspaces, which can be
computed as lengths of determinantal modules. In addition to being applied to
symmetric, general and skew-symmetric matrix singularities, the results are
also applied to compute the Milnor number of isolated Cohen-Macaulay surface
singularities in $\C^4$ and the difference of Betti numbers of Milnor fibers
for isolated Cohen-Macaulay 3-fold singularities in $\C^5$.

This paper is the first part of a two part paper which introduces a method
for determining the vanishing topology of nonisolated matrix singularities. A
foundation for this is the introduction of a method for obtaining new classes
of free divisors from representations $V$ of connected solvable linear
algebraic groups $G$. For equidimensional representations where $\dim G=\dim
V$, with $V$ having an open orbit, we give sufficient conditions that the
complement $E$ of this open orbit, the "exceptional orbit variety", is a free
divisor (or a weaker free* divisor).
We do so by introducing the notion of a "block representation" which is
especially suited for both solvable groups and extensions of reductive groups
by them. This is a representation for which the matrix representing a basis of
associated vector fields on $V$ defined by the representation can be expressed
using a basis for $V$ as a block triangular matrix, with the blocks satisfying
certain nonsingularity conditions. We use the Lie algebra structure of $G$ to
identify the blocks, the singular structure, and a defining equation for $E$.
This construction naturally fits within the framework of towers of Lie groups
and representations yielding a tower of free divisors which allows us to
inductively represent the variety of singular matrices as fitting between two
free divisors. We specifically apply this to spaces of matrices including
$m\times m$ symmetric, skew-symmetric or general matrices, where we prove that
both the classical Cholesky factorization of matrices and a "modified Cholesky
factorization" which we introduce are given by block representations of
solvable group actions. For skew-symmetric matrices, we use an extension of the
method on an infinite dimensional solvable Lie algebra.
In part II, we use these results to compute the vanishing topology for matrix
singularities for all of the classes.

Every real skew-symmetric matrix B admits Cholesky-like factorizations B = R T JR, where J = # 0 -I I 0 # . This paper presents a backward-stable O(n 3 ) process for computing such a decomposition, in which R is a permuted triangular matrix. Decompositions of this type are a key ingredient of algorithms for solving eigenvalue problems with Hamiltonian structure.

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 provide a criterion that for an equivalence group G on holomorphic germs, the discriminant of a G-versal unfolding is a free divisor. The criterion is in terms of the discriminant being Cohen-Macaulay and generically having Morse-type singularities. When either of these conditions fails, we provide a criterion that the discriminant have a weaker free* divisor structure. For nonlinear sections of a free* divisor V, we obtain a formula for the number of singular vanishing cycles by modifying an earlier formula obtained with David Mond and taking into account virtual singularities.

Preface to the Second Edition Preface to the First Edition List of Figures Prerequisites by Chapter Standard Notation Introduction: Closed Linear Groups Lie Algebras and Lie Groups Complex Semisimple Lie Algebras Universal Enveloping Algebra Compact Lie Groups Finite-Dimensional Representations Structure Theory of Semisimple Groups Advanced Structure Theory Integration Induced Representations and Branching Theorems Prehomogeneous Vector Spaces Appendices Hints for Solutions of Problems Historical Notes References Index of Notation Index

Suppose that f: C(n), 0 --> C(p), 0 is finitely A-determined with n greater-than-or-equal-to p. We define a "Milnor fiber" for the discriminant of f; it is the discriminant of a "stabilization" of f. We prove that this "discriminant Milnor fiber" has the homotopy type of a wedge of spheres of dimension p - 1, whose number we denote by mu-DELTA(f). One of the main theorems of the paper is a "mu = tau" type result: if (n, p) is in the range of nice dimensions in the sense of Mather, then mu-DELTA(f) greater-than-or-equal-to A(e)-codim(f), with equality if f is weighted homogeneous. Outside the nice dimensions we obtain analogous formulae with correction terms measuring the presence of unstable but topologically stable germs in the stabilization. These results are further extended to nonlinear sections of free divisors.

Let G be a connected linear algebraic group, and p a rational representation of G on a finite-dimensional vector space V , all defined over the complex number field C .
We call such a triplet ( G, p, V ) a prehomogeneous vector space if V has a Zariski-dense G -orbit. The main purpose of this paper is to classify all prehomogeneous vector spaces when p is irreducible, and to investigate their relative invariants and the regularity.

Let V be a complex vector space of dimension l and let G ⊂ GL(V ) be a finite reflection group. Let S be the C -algebra of polynomial functions on V with its usual G -module structure ( gf)(v ) = f{g ⁻¹ v ). Let R be the subalgebra of G -invariant polynomials. By Chevalley’s theorem there exists a set ℬ = { f 1 , …, f l } of homogeneous polynomials such that R = C[f 1 , …, f l ]. We call ℬ a set of basic invariants or a basic set for G . The degrees d i = deg f i are uniquely determined by G . We agree to number them so that d 1 ≤ … ≤ d i . The map τ : V/G → C ¹ defined by
is a bijection. Each reflection in G fixes some hyperplane in V .

Contenido: Introducción; Resolución de ecuaciones lineales; Problemas menos lineales de cuadrados; Problemas de valor propio no simétricos; El problema de valor propio simétrico y el valor singular de descomposición; Métodos iterativos para sistemas lineales; Métodos iterativos para problemas de valor propio.

Sur les Groupes de Tresses (aprés Arnold), in: Séminaire Bourbaki

- E V Brieskorn

E.V. Brieskorn, Sur les Groupes de Tresses (aprés Arnold), in: Séminaire Bourbaki 1971/1972, in: Springer Lecture Notes in Math., vol. 317, 1973.

On the legacy of free divisors II: Free * divisors and complete intersections, in: Special Issue in Honor of V.I. Arnol'd, Mosc

- J Damon

J. Damon, On the legacy of free divisors II: Free * divisors and complete intersections, in: Special Issue in Honor of V.I. Arnol'd, Mosc. Math. J. 3 (2) (2003) 361–395.

Linear Free Divisors and Quiver Representations in Singularities and Computer Algebra London math

- R O Buchweitz
- D Mond

Buchweitz, R. O. and Mond, D. Linear Free Divisors and Quiver Representations in
Singularities and Computer Algebra London math. Soc. Lect. Ser. vol 324 Cambridge
Univ. Press, 2006, 41-77 6

Sur les Groupes de Tresses (aprés Arnold) Séminaire Bourbaki (1971/72) Springer Lect

- E V Brieskorn

Brieskorn, E. V. Sur les Groupes de Tresses (aprés Arnold) Séminaire Bourbaki
(1971/72) Springer Lect. Notes in Math. 317, (1973) 1