The inverse of a quantum bilinear form of the oriented braid arrangement

Abstract

We follow here the results of Varchenko, who assigned to each weighted arrangement \(\mathcal{A}\) of hyperplanes in the \(n\)-dimensional real space a bilinear form, which he called the quantum bilinear form of the arrangement \(\mathcal{A}\). We briefly explain the quantum bilinear form of the oriented braid arrangement in the \(n\)-dimensional real space. The main concern of this paper is to compute the inverse of the matrix of the quantum bilinear form of the oriented braid arrangement in \(\mathbb{R}^n\), \({n\ge 2}\). To solve this problem, in [3] the authors used some special matrices and their factorizations in terms of simpler matrices. So, to simplify some matrix calculations, we first introduce a twisted group algebra \({\mathcal{A}(S_{n})}\) of the symmetric group \(S_{n}\) with coefficients in the polynomial ring in \(n^2\) commutative variables and then use a natural representation of some elements of the algebra \({\mathcal{A}(S_{n})}\) on the generic weight subspaces of the multiparametric quon algebra \({\mathcal{B}}\), which immediately gives the corresponding matrices of the quantum bilinear form.