Glasnik Matematicki, Vol. 59, No. 1 (2024), 5175. \( \)
THE INVERSE OF A QUANTUM BILINEAR FORM OF THE ORIENTED BRAID ARRANGEMENT
Milena Sošić
Faculty of Mathematics, University of Rijeka, 51 000 Rijeka, Croatia
email: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
Full text (PDF) (access from subscribing institutions only)
https://doi.org/10.3336/gm.59.1.03
References:

T. Brylawski and A. Varchenko, The determinant formula for a matroid bilinear form, Adv. Math. 129 (1997), 1–24.
MathSciNet
CrossRef

G. Duchamp, A. Klyachko, D. Krob and J.Y. Thibon, Noncommutative symmetric functions III: Deformations of Cauchy and convolution algebras, Discrete Math. Theor. Comput. Sci. 1 (1997), 159–216.
MathSciNet
CrossRef

S. Meljanac and D. Svrtan, Study of Gram matrices in Fock representation of multiparametric canonical commutation relations, extended Zagier's conjecture, hyperplane arrangements and quantum groups, Math. Commun. 1 (1996), 1–24.
MathSciNet

P. Orlik and H. Terao, Arrangements of hyperplanes, SpringerVerlag, Berlin, 1992.
MathSciNet
CrossRef

M. Sošić, A representation of twisted group algebra of symmetric groups on weight subspaces of free associative complex algebra, Math. Forum 26 (2014), 23–48.
MathSciNet

M. Sošić, Computation of constants in multiparametric quon algebras. A twisted group algebra approach, Math. Commun. 22, (2017), 177–192.
MathSciNet

M. Sošić, On the Varchenko determinant formula for oriented braid arrangements, Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 27 (2023), 203218.
MathSciNet
CrossRef

M. Sošić, Some factorizations in the twisted group algebra of symmetric groups, Glas. Mat. Ser. III 51(71), (2016) 1–15.
MathSciNet
CrossRef

D. Stanton and D. White, Constructive Combinatorics, SpringerVerlag, New York, 1986.
MathSciNet
CrossRef

A. Varchenko, Bilinear form of real configuration of hyperplanes, Adv. Math. 97 (1993), 110–144.
MathSciNet
CrossRef
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