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The Varchenko determinant of a Coxeter arrangement. (English) Zbl 1403.20054
An arrangement of hyperplanes in \(\mathbb{R}^n\) is a finite set of hyperplanes. A chamber of a hyperplane arrangement \(\mathcal{A}\) is a connected component of the complement \(\mathbb{R}^n \backslash \bigcup_{H \in \mathcal{A}} H\). The Varchenko determinant is the determinant of a matrix defined from an arrangement of hyperplanes.
An edge of a hyperplane arrangement \(\mathcal{A}\) is a nonempty intersection of some of its hyperplanes. Denote the set of all edges of \(\mathcal{A}\) by \(L(\mathcal{A})\). The weight a(\(E\)) of an edge \(E\) is a\((E) := \prod a_H\), where \(H\in \mathcal{A}\), \(a_H\) is the weight of \(H\) and \(E\subseteq H\). The multiplicity \(l(E)\) of an edge \(E\) is a positive integer.
The hyperplane arrangement associated to a finite Coxeter group is called a Coxeter arrangement. Let \(H_t\) be the hyperplane \(\text{ker}(t - 1)\) of \(\mathbb{R}^n\) whose points are fixed by each element \(t\) of \(T\), the set of all reflections of the finite Coxeter group \(W\). Let \(S\) be the set of simple reflections of \(W\) and let \(J\) be a subset of \(S\). The hyperplane arrangement associated to \(W\) is \(\mathcal{A}_W := \{H_t\}_{t \in T}\). Let \(E\) be an edge of \(\mathcal{A}_W\). The main result that the authors obtain are formulas for a\((E)\) and \(l(E)\): \[ a(E_{T_J^w}) = \prod_{u \in T_J}a_{H_u}w, \quad l(E_{T_J^w})= | \lfloor t_J\rceil | \cdot|\left[J\right]|\cdot|X(S,J)|\cdot|X(J,\{s_J\})|. \]
Applying this result, the authors compute the Varchenko determinant of a finite Coxeter group. They use Table 1: Number of full support reflections of the irreducible finite Coxeter groups, and Tables 2 and 3: Multiplicities of the Coxeter classes for all irreducible finite Coxeter groups.

20F55 Reflection and Coxeter groups (group-theoretic aspects)
51F15 Reflection groups, reflection geometries
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
05E10 Combinatorial aspects of representation theory
Full Text: DOI arXiv
[1] P. Abramenko and K. S. Brown, Buildings. Theory and Applications, Grad. Texts in Math. 248, Springer, New York, 2008.
[2] A. Björner and F. Brenti, Combinatorics of Coxeter Groups, Grad. Texts in Math. 231, Springer, New York, 2005.
[3] N. Bourbaki, Éléments de mathématique. Groupes et algèbres de Lie. Chapitres 4, 5 et 6, Masson, Paris, 1981.
[4] H. S. M. Coxeter, Discrete groups generated by reflections, Ann. of Math. (2) 35 (1934), no. 3, 588-621. · JFM 60.0898.02
[5] G. Denham and P. Hanlon, Some algebraic properties of the Schechtman-Varchenko bilinear forms, New Perspectives in Algebraic Combinatorics (Berkeley 1996/97), Math. Sci. Res. Inst. Publ. 38, Cambridge University Press, Cambridge (1999), 149-176. · Zbl 1050.52501
[6] 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), no. 1, 159-216. · Zbl 0930.05097
[7] M. Geck and G. Pfeiffer, Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras, London Math. Soc. Monogr. (N. S.) 21, Oxford University Press, New York, 2000. · Zbl 0996.20004
[8] R. B. Howlett, Normalizers of parabolic subgroups of reflection groups, J. Lond. Math. Soc. (2) 21 (1980), no. 1, 62-80. · Zbl 0427.20040
[9] J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Stud. Adv. Math. 29, Cambridge University Press, Cambridge, 1990.
[10] H. Randriamaro, Computing the Varchenko determinant of a bilinear form, preprint (2014), .
[11] V. V. Schechtman and A. N. Varchenko, Quantum groups and homology of local systems, Algebraic Geometry and Analytic Geometry (Tokyo 1990), ICM-90 Satell. Conf. Proc., Springer, Tokyo (1991), 182-197. · Zbl 0760.17014
[12] A. Varchenko, Bilinear form of real configuration of hyperplanes, Adv. Math. 97 (1993), no. 1, 110-144. · Zbl 0777.52006
[13] D. Zagier, Realizability of a model in infinite statistics, Comm. Math. Phys. 147 (1992), no. 1, 199-210. · Zbl 0789.47042
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