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On the invariants of the cohomology of complements of Coxeter arrangements. (English) Zbl 1455.20003
Let $$W$$ be a finite Coxeter group with a (Coxeter) generating set $$S$$ of order $$\ell$$, $$V_\mathbb{R}$$ be an $$\ell$$-dimensional real vector space that affords the reflection representation of $$W$$, $$V= \mathbb{C} \otimes_{\mathbb{R}} V_{\mathbb{R}}$$ be the complexification of $$V_{\mathbb{R}}$$ and let $$R$$ be the set of reflections of $$W$$. If $$W$$ is seen as a subgroup $$\mathrm{GL}(V)$$, for every $$r\in R$$ let $$V^{r}$$ be the hyperplane of fixed points of $$r$$ in $$V$$ and let $$\mathcal{A} =\{ V^{r} \mid r \in R \}$$. Then $$(V, \mathcal{A})$$ is the complexification of a Coxeter arrangement.
The group $$W$$ acts naturally on the complement $$M_{W}$$ of the hyperplanes in $$\mathcal{A}$$, and hence on the cohomology of $$M_{W}$$ as algebra automorphisms. For $$p \geq 0$$ let $$H^{p}(M_{W})$$ denote the $$p^{\mathrm{th}}$$ de Rahm cohomology space of $$M_{W}$$.
G. Felder and A. P. Veselov [J. Eur. Math. Soc. (JEMS) 7, No. 1, 101–116 (2005; Zbl 1070.20045)] have conjectured an explicit construction of $$H^{p}(M_{W})^{W}$$, the space of $$W$$-invariants in $$H^{p} (M_{W})$$, in terms of so-called special involutions and they have verified their conjecture for all Coxeter groups except those with irreducible components of type $$E_{7}$$, $$E_{8}$$, $$F_{4}$$, $$H_{3}$$ or $$H_{4}$$.
The purpose of the paper under review is to complete the proof of the conjecture of Felder-Veselov (Theorem 2.1). The authors get their result by refining E. Brieskorn’s [Lect. Notes Math. 317, 21–44 (1973; Zbl 0277.55003)] study of the cohomology of the complement of the reflection arrangement of the Coxeter group $$W$$.
##### MSC:
 20C08 Hecke algebras and their representations 20F55 Reflection and Coxeter groups (group-theoretic aspects) 05E10 Combinatorial aspects of representation theory 52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
CHEVIE; GAP
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##### References:
 [1] Bishop, M.; Douglass, J. M.; Pfeiffer, G.; Röhrle, G., Computations for Coxeter arrangements and Solomon’s descent algebra: groups of rank three and four, J. Symbolic Comput., 50, 139-158 (2013) · Zbl 1257.20004 [2] Bishop, M.; Douglass, J. M.; Pfeiffer, G.; Röhrle, G., Computations for Coxeter arrangements and Solomon’s descent algebra II: groups of rank five and six, J. Algebra, 377, 320-332 (2013) · Zbl 1277.20007 [3] Bishop, M.; Douglass, J. M.; Pfeiffer, G.; Röhrle, G., Computations for Coxeter arrangements and Solomon’s descent algebra III: groups of rank seven and eight, J. Algebra, 423, 1213-1232 (2015) · Zbl 1306.20004 [4] Brieskorn, E., Sur les groupes de tresses [d’après V.I. Arnold], (Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 401. Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 401, Lecture Notes in Math., vol. 317 (1973), Springer: Springer Berlin), 21-44 [5] Felder, G.; Veselov, A. P., Coxeter group actions on the complement of hyperplanes and special involutions, J. Eur. Math. Soc. (JEMS), 7, 1, 101-116 (2005) · Zbl 1070.20045 [6] Geck, M.; Hiß, G.; Lübeck, F.; Malle, G.; Pfeiffer, G., CHEVIE — a system for computing and processing generic character tables, Appl. Algebra Engrg. Comm. Comput., 7, 175-210 (1996) · Zbl 0847.20006 [7] Geck, M.; Pfeiffer, G., Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras, London Mathematical Society Monographs. New Series, vol. 21 (2000), The Clarendon Press, Oxford University Press: The Clarendon Press, Oxford University Press New York · Zbl 0996.20004 [8] Howlett, R. B., Normalizers of parabolic subgroups of reflection groups, J. Lond. Math. Soc. (2), 21, 1, 62-80 (1980) · Zbl 0427.20040 [9] Lehrer, G. I.; Solomon, L., On the action of the symmetric group on the cohomology of the complement of its reflecting hyperplanes, J. Algebra, 104, 2, 410-424 (1986) · Zbl 0608.20010 [10] Richardson, R. W., Conjugacy classes of involutions in Coxeter groups, Bull. Aust. Math. Soc., 26, 1, 1-15 (1982) · Zbl 0531.20017 [11] Orlik, P.; Solomon, L., Combinatorics and topology of complements of hyperplanes, Invent. Math., 56, 2, 167-189 (1980) · Zbl 0432.14016 [12] Orlik, P.; Terao, H., Arrangements of Hyperplanes (1992), Springer-Verlag · Zbl 0757.55001 [13] Pfeiffer, G.; Röhrle, G., Special involutions and bulky parabolic subgroups in finite Coxeter groups, J. Aust. Math. Soc., 79, 1, 141-147 (2005) · Zbl 1078.20042 [14] Schönert, M., GAP - Groups, Algorithms, and Programming - Version 3 Release 4 (1997), Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule: Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule Aachen, Germany
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