Publication date: 20 June 2024
Volume: Vol 59
Issue: Svezak 1
Pages: 51-75
DOI: 10.3336/gm.59.1.03
Izvorni znanstveni članak
https://doi.org/10.3336/gm.59.1.03
The inverse of a quantum bilinear form of the oriented braid arrangement
Milena Sošić
orcid.org/0000-0002-0085-2073
; Faculty of Mathematics, University of Rijeka, 51 000 Rijeka, Croatia
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.
Oriented braid arrangement, twisted group algebra, multiparametric quon algebra
318150
16.8.2024.
Posjeta: 71 *