Free hyperplane arrangements between \(A_{n-1}\) and \(B_ n\)
Much of the motivation for the study of arrangements of hyperplanes comes from the study of Coxeter arrangements. The sub-arrangements of Coxeter arrangements are less well understood. In the case of the root system \(A_{n-1}\) the sub-arrangements are the graphic arrangements, and using this connection to graphs one can analyze these arrangements completely from the perspective of freeness (e.g. Theorem 3.3). \textit{T. Zaslavsky} has developed a theory of signed graphs [Am. Math. Mon. 88, 88-105 (1981; Zbl 0466.05058)] which, in principle, should allow one to analyze the sub-arrangements of \(B_ n\) in a similar way. No one has yet been able to put that into practice. For the class of sub-arrangements of \(B_ n\) that contain \(A_{n-1}\) there is again a natural connection to graphs. It is this class of arrangements that we study. In this paper we describe completely which of these arrangements are free in a manner analogous to the situation for sub-arrangements of \(A_{n-1}\). We were partly motivated by the recent work of \textit{T. Józefiak} and \textit{B. E. Sagan} [J. Algebraic Combinatorics 2, 291-320 (1993)] on a certain subclass of these free arrangements. The paper is organized as follows. In the next section we collect the necessary known results concerning arrangements and establish graph- theoretic terminology. In \S3 we characterize the sub-arrangements of \(A_{n-1}\) that are free, namely those parameterized by chordal graphs. For the most part these results are known, but they are not collected anywhere. The techniques employed in this section also serve to foreshadow those in \S4. In \S4 we classify the sub-arrangements of \(B_ n\) that contain \(A_{n-1}\) and are free. They are parameterized by threshold graphs. We also show which of these free arrangements are supersolvable. A by-product of our analysis is an infinite collection of arrangements that are counterexamples to Orlik's conjecture. In addition, we are able to enumerate all of the induction tables from \(A_{n-1}\) to \(B_ n\).
scientific article; zbMATH DE number 567407
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