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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.

##### MSC:
 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
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##### References:
 [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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