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

We follow here the results of Varchenko, who assigned to each weighted arrangement \(A\) of hyperplanes in the \(n\)-dimensional real space a bilinear form, which he called the quantum bilinear form of the arrangement \(A\). 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 \(R^n\), \({n≥ 2}\). 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 \({A(Sₙ)}\) of the symmetric group \(Sₙ\) with coefficients in the polynomial ring in \(n^2\) commutative variables and then use a natural representation of some elements of the algebra \({A(Sₙ)}\) on the generic weight subspaces of the multiparametric quon algebra \({B}\), which immediately gives the corresponding matrices of the quantum bilinear form.