Rad HAZU, Matematičke znanosti, Vol. 27 (2023), 203-218.

ON THE VARCHENKO DETERMINANT FORMULA FOR ORIENTED BRAID ARRANGEMENTS

Milena Sošić

Abstract.   In this paper, we first consider the arrangement of hyperplanes and then the corresponding oriented arrangement of hyperplanes in n-dimensional real space. Following the work of Varchenko, who studied the determinant of the quantum bilinear form of a real configuration and the determinant formula for a matroid bilinear form, we discuss here first some of the main properties of the braid arrangement and then of the oriented braid arrangements in n-dimensional real space. The main result of this study is a theorem that provides an explicit formula for determining the determinant of the matrix associated with the oriented braid arrangement. The proof of this theorem is based on the results of two different approaches. One is to determine the space of all constants in the multiparametric quon algebra equipped with a multiparametric q-differential structure, and the other is to study the feasibility of multiparametric quon algebras in Hilbert space.

2020 Mathematics Subject Classification.   52C35, 05E16.

Key words and phrases.   Arrangements of hyperplanes, braid arrangement, oriented braid arrangements, quantum bilinear form, multiparametric quon algebra.

DOI: https://doi.org/10.21857/moxpjh1l0m

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