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167
Complex reflection groups , Braid groups, Hecke algebras
, 1997
"... Presentations "a la Coxeter" are given for all (irreducible) finite complex reflection groups. They provide presentations for the corresponding generalized braid groups (for all but six cases), which allow us to generalize some of the known properties of finite Coxeter groups and their associated ..."
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Cited by 118 (9 self)
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Presentations "a la Coxeter" are given for all (irreducible) finite complex reflection groups. They provide presentations for the corresponding generalized braid groups (for all but six cases), which allow us to generalize some of the known properties of finite Coxeter groups and their associated braid groups, such as the computation of the center of the braid group and the construction of deformations of the finite group algebra (Hecke algebras). We introduce monodromy representations of the braid groups which factorize through the Hecke algebras, extending results of Cherednik, Opdam, Kohno and others.
Arrangements and Their Applications
 Handbook of Computational Geometry
, 1998
"... The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arr ..."
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Cited by 75 (20 self)
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The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization. Some results involving planar arrangements of arcs have been presented in a companion chapter in this book, and are extended in this chapter to higher dimensions. Work by P.A. was supported by Army Research Office MURI grant DAAH049610013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.Israeli Binational Science Foundation. Work by M.S. was supported by NSF Grants CCR9122103 and CCR9311127, by a MaxPlanck Research Award, and by grants from the U.S.Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Ac...
Characteristic varieties of arrangements
 MATH. PROC. CAMBRIDGE PHIL. SOC. 127 (1999), 33–53. MR 2000M:32036
, 1999
"... Abstract. The kth Fitting ideal of the Alexander invariant B of an arrangement A of n complex hyperplanes defines a characteristic subvariety, Vk(A), of the algebraic torus (C∗) n. In the combinatorially determined case where B decomposes as a direct sum of local Alexander invariants, we obtain a co ..."
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Cited by 64 (17 self)
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Abstract. The kth Fitting ideal of the Alexander invariant B of an arrangement A of n complex hyperplanes defines a characteristic subvariety, Vk(A), of the algebraic torus (C∗) n. In the combinatorially determined case where B decomposes as a direct sum of local Alexander invariants, we obtain a complete description of Vk(A). For any arrangement A, we show that the tangent cone at the identity of this variety coincides with R1 k (A), one of the cohomology support loci of the OrlikSolomon algebra. Using work of Arapura [1] and Libgober [18], we conclude that all positivedimensional components of Vk(A) are combinatorially determined, and that R1 k (A) is the union of a subspace arrangement in Cn, thereby resolving a conjecture of Falk [11]. We use these results to study the reflection arrangements associated to monomial groups.
Coxeter arrangements
 Proceedings of Symposia in Pure Mathematics 40
, 1983
"... Let V be an ℓdimensional Euclidean space. Let G ⊂ O(V) be a finite irreducible orthogonal reflection group. Let A be the corresponding Coxeter arrangement. Let S be the algebra of polynomial functions on V. For H ∈ A choose αH ∈ V ∗ such that H = ker(αH). For each nonnegative integer m, define the ..."
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Cited by 60 (6 self)
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Let V be an ℓdimensional Euclidean space. Let G ⊂ O(V) be a finite irreducible orthogonal reflection group. Let A be the corresponding Coxeter arrangement. Let S be the algebra of polynomial functions on V. For H ∈ A choose αH ∈ V ∗ such that H = ker(αH). For each nonnegative integer m, define the derivation module D (m) (A) = {θ ∈ DerS  θ(αH) ∈ Sα m H}. The module is known to be a free Smodule of rank ℓ by K. Saito (1975) for m = 1 and L. SolomonH. Terao (1998) for m = 2. The main result of this paper is that this is the case for all m. Moreover we explicitly construct a basis for D (m) (A). Their degrees are all equal to mh/2 (when m is even) or are equal to ((m − 1)h/2) + mi(1 ≤ i ≤ ℓ) (when m is odd). Here m1 ≤ · · · ≤ mℓ are the exponents of G and h = mℓ + 1 is the Coxeter number. The construction heavily uses the primitive derivation D which plays a central role in the theory of flat generators by K. Saito (or equivalently the Frobenius manifold structure for the orbit space of G.) Some new results concerning the primitive derivation D are obtained in the course of proof of the main result.
ARRANGEMENTS AND LOCAL SYSTEMS
 MATHEMATICAL RESEARCH LETTERS 7, 299–316 (2000)
, 2000
"... We use stratified Morse theory to construct a complex to compute the cohomology of the complement of a hyperplane arrangement with coefficients in a complex rank one local system. The linearization of this complex is shown to be the Aomoto complex of the arrangement. Using this result, we establish ..."
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Cited by 47 (8 self)
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We use stratified Morse theory to construct a complex to compute the cohomology of the complement of a hyperplane arrangement with coefficients in a complex rank one local system. The linearization of this complex is shown to be the Aomoto complex of the arrangement. Using this result, we establish the relationship between the cohomology support loci of the complement and the resonance varieties of the OrlikSolomon algebra for any arrangement, and show that the latter are unions of subspace arrangements in general, resolving a conjecture of Falk. We also obtain lower bounds for the local system Betti numbers in terms of those of the OrlikSolomon algebra, recovering a result of Libgober and Yuzvinsky. For certain local systems, our results provide new combinatorial upper bounds on the local system Betti numbers. These upper bounds enable us to prove that in nonresonant systems the cohomology is concentrated in the top dimension, without using resolution of singularities.
Fundamental groups of line arrangements: Enumerative aspects
 Contemporary Mathematics
, 2001
"... This is a survey of some recent developments in the study of complements of line arrangements in the complex plane. We investigate the fundamental groups and finite covers of those complements, focusing on homological and enumerative aspects. The unifying framework for this study is the stratificati ..."
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Cited by 46 (14 self)
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This is a survey of some recent developments in the study of complements of line arrangements in the complex plane. We investigate the fundamental groups and finite covers of those complements, focusing on homological and enumerative aspects. The unifying framework for this study is the stratification of the character variety of the fundamental group, G, by the jumping loci for cohomology with coefficients in rank 1 local systems. Counting certain torsion points on these "characteristic" varieties yields information about the homology of branched and unbranched covers of the complement, as well as on the number of lowindex subgroups of its fundamental group. We conclude with two conjectures, expressing the lower central series quotients of G/G'' (and, in some cases, G itself) in terms of the closely related "resonance" varieties. We illustrate the discussion with a number of detailed examples, some of which reveal new phenomena.
Translated tori in the characteristic varieties of complex hyperplane arrangements
 TOPOLOGY AND ITS APPLICATIONS
, 2002
"... We give examples of complex hyperplane arrangements for which the top characteristic variety contains positivedimensional irreducible components that do not pass through the origin of the character torus. These examples answer several questions of Libgober and Yuzvinsky. As an application, we exhib ..."
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Cited by 41 (12 self)
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We give examples of complex hyperplane arrangements for which the top characteristic variety contains positivedimensional irreducible components that do not pass through the origin of the character torus. These examples answer several questions of Libgober and Yuzvinsky. As an application, we exhibit a pair of arrangements for which the resonance varieties of the OrlikSolomon algebra are (abstractly) isomorphic, yet whose characteristic varieties are not isomorphic. The difference comes from translated components, which are not detected by the tangent cone at the origin.
Hypersurface complements, Milnor fibers and higher homotopy groups of arrangments
 Ann. of Math
"... The interplay between geometry and topology on complex algebraic varieties is a classical theme that goes back to Lefschetz [L] and Zariski [Z] and is always present on the scene, see for instance the work by Libgober [Li]. In this paper we study complements of hypersurfaces, with a special attentio ..."
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Cited by 38 (8 self)
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The interplay between geometry and topology on complex algebraic varieties is a classical theme that goes back to Lefschetz [L] and Zariski [Z] and is always present on the scene, see for instance the work by Libgober [Li]. In this paper we study complements of hypersurfaces, with a special attention to the case of hyperplane arrangements as discussed
On the fundamental group of the complement of a complex hyperplane arrangement
, 1994
"... Abstract. We construct two combinatorially equivalent line arrangements in the complex projective plane such that the fundamental groups of their complements are not isomorphic. In the proof we use a new invariant of the fundamental group of the complement of a line arrangement with prescribed combi ..."
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Cited by 32 (1 self)
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Abstract. We construct two combinatorially equivalent line arrangements in the complex projective plane such that the fundamental groups of their complements are not isomorphic. In the proof we use a new invariant of the fundamental group of the complement of a line arrangement with prescribed combinatorial type with respect to isomorphisms inducing the canonical isomorphism of first homology groups. 1.