Bilinear form of real configuration of hyperplanes
A real configuration \({\mathcal C}\) is a finite set of hyperplanes in some real affine space (synonymous to real arrangement of hyperplanes in \textit{P. Orlik} and \textit{H. Terao} [Arrangements of hyperplanes (1992; Zbl 0757.55001)], any nonempty intersection of some subset of \({\mathcal C}\) is called an edge, the connected components of the complement of the union of \({\mathcal C}\) are called domains. If to any hyperplane \(H\in{\mathcal C}\) there is assigned a real number \(a(H)\) called weight, then edges are weighted with the product of all hyperplanes containing them. Let \(A_{\mathcal C}\) be the ring of polynomials with \(\{a(H)| H\in{\mathcal C}\}\) as set of variables, and with integer coefficients, \(M_{\mathcal C}\) the space of \(A_{\mathcal C}\)- linear combinations of domains of \({\mathcal C}\). Then \(B(P,Q):=\prod\{a(H)| H\in{\mathcal C}\) separates \(P\) and \(Q\}\) defines an \(A_{\mathcal C}\)-bilinear symmetric form in \(M_{\mathcal C}\), which the author calls the quantum bilinear form of \({\mathcal C}\). The major part of the paper is devoted to the proof of the following. Theorem. The determinant of \(B\) is equal to a product of powers of binomials of the form \(1-a(L)^ 2\) where \(L\) runs through all the edges of \({\mathcal C}\), and any such binomial occurs with an exponent which is shown to be the product of the number of domains of the configuration in \(L\) induced by restriction of \({\mathcal C}\), and of some well-defined integer which depends on how \({\mathcal C}\) restricts to an arbitrarily chosen normal of \(L\). In particular, if \(B\) belongs to the configuration of all hyperplanes \(\text{Ker}(t_ i-t_ j)\) in real \(k\)-space where \(1\leq i\leq j\leq k\), then it can be interpreted as the contravariant form of some suitable quantum group [cf. \textit{V. Schechtman} and the author, ICM-90 Satell. Conf. Proc., 182-197 (1991; Zbl 0760.17014)].
scientific article; zbMATH DE number 221227