Glasnik Matematicki, Vol. 59, No. 1 (2024), 51-75. \( \)

THE INVERSE OF A QUANTUM BILINEAR FORM OF THE ORIENTED BRAID ARRANGEMENT

Milena Sošić

Faculty of Mathematics, University of Rijeka, 51 000 Rijeka, Croatia
e-mail:msosic@uniri.hr

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.

2020 Mathematics Subject Classification.   16S32, 05E16, 52C35

Key words and phrases.   Oriented braid arrangement, twisted group algebra, multiparametric quon algebra

https://doi.org/10.3336/gm.59.1.03

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