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Solvable group representations and free divisors whose complements are K(π, 1)'s

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Abstract

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]. ...
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... 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). ...
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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. ...
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Preprint
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.
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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
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Bibliogr. s. 303-314
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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 .
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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
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On the legacy of free divisors II: Free * divisors and complete intersections, in: Special Issue in Honor of V.I. Arnol'd, Mosc
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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
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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
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