A classical result of Arnold and Brieskorn states that the complement of the discriminant of the versal unfolding of a simple hypersurface singularity is a \(K(\pi,1)\). Deligne showed this result could be placed in the general framework by proving that the complement of an arrangement of reflecting hyperplanes for a Coxeter group is again a \(K(\pi, 1)\). What the discriminants and Coxeter hyperplane arrangements have in common is that they are free divisors. This notion was introduced by Saito. This leads to an intriguing question about when a free divisor has a complement which is a \(K(\pi,1)\). This remains unsettled for the discriminants of versal unfoldings of isolated hypersurface singularities; this is the classical “\(K(\pi,1)\)-problem”. Also, it remains open whether the conjecture of Saito is true that every free arrangement has a complement which is a \(K(\pi,1)\). While neither \(K(\pi,1)\)-problem has been settled, numerous other classes of free divisors have been discovered so this question continues to arise in new contexts.
In this paper, the authors 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.