Combinatorics of inductively factored arrangements
As a part of the continuing effort to explain the ``miraculous'' factorizations of polynomials associated with certain arrangements of hyperplanes, \textit{H. Terao} [Adv. Math. 91, 45-53 (1992; Zbl 0772.05029)] had introduced the class of factored hyperplane arrangements, which generalizes the class of supersolvable arrangements: an arrangement is factored if its set of hyperplanes admits a partition that induces a complete factorization of the Orlik-Solomon algebra as a \(Z\)-module. In this paper, the authors introduce a class of inductively factored arrangements, that is, factored arrangements with nice recursive properties. They establish a hierarchy ``supersolvable \(\Rightarrow\) inductively factored \(\Rightarrow\) hyperfactored \(\Rightarrow\) factored'', where a (real) arrangement is hyperfactored if there is a rank and order preserving bijection from a product of chains to its poset of regions (with respect to a suitable base region). The main results of this paper thus establish a generalization of main results of \textit{A. Björner, P. Edelman} and the reviewer [Discrete Comput. Geom. 5, No. 3, 263-288 (1990; Zbl 0698.51010)] from supersolvable arrangements to the somewhat larger class of inductively factored arrangements. The most interesting part of the paper may be an interpretation of the ``hyperfactored'' property as providing a subdivision of the boundary of a cube of appropriate dimension into cubic cells. It is also proved that inductively factored arrangements are free in the sense of \textit{M. Terao} [J. Fac. Sci., Univ. Tolyo, Sect. I A 27, 293-312 (1980; Zbl 0509.14006)].
scientific article; zbMATH DE number 773831
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