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      Solvable Group Representations and Free Divisors whose Complements are \(K(\pi, 1)\)'s



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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 \(K(\pi, 1)\)'s. 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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          A classification of irreducible prehomogeneous vector spaces and their relative invariants

           M Sato,  E T Kimura (1977)
          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.
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            Linear free divisors and the global logarithmic comparison theorem

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              Discriminants in the invariant theory of reflection groups

              LetVbe a complex vector space of dimensionland letG⊂GL(V) be a finite reflection group. LetSbe theC-algebra of polynomial functions onVwith its usualG-module structure (gf)(v) =f{g-1v). LetRbe the subalgebra ofG-invariant polynomials. By Chevalley’s theorem there exists a setℬ= {f1, …,fl} of homogeneous polynomials such thatR=C[f1, …,fl]. We callℬa set of basic invariants or abasic setforG. The degreesdi= degfiare uniquely determined byG. We agree to number them so thatd1≤ … ≤di. The mapτ:V/G → C1defined by

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                30 October 2013


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                22E25, 55P20, 55R80 (Primary) 20G05, 32S30 (Secondary)
                Topology Appl. 159 (2012), no. 2, 437--449
                19 pages, pdflatex. Reviewed as MR2868903, Zbl 1257.55010. NOTICE: This is the author's version of a work that was accepted for publication in Topology and its Applications. Changes resulting from the publishing process may not be reflected in this document. A definitive version was subsequently published in Topology Appl. 159 (2012), no. 2, 437--449
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