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

### Most cited references3

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(1988)
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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### Author and article information

###### Journal
30 October 2013
###### Article
10.1016/j.topol.2011.09.018
1310.8280