Homology computations for complex braid groups

Callegaro, Filippo; Marin, Ivan

doi:10.4171/jems/429

Cited by 17 publications

(38 citation statements)

References 24 publications

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“…By several arguments, recalled in [5], one can reduce the problem to a fewer number of groups. In particular, the homology of groups of small rank can be easily computed.…”

Section: Introductionmentioning

confidence: 99%

“…The drawback of this complex is that, while the computation of the homology of the complex is much easier as soon as it is explicitely described, the explicit computation of (the differential of) the complex itself is much more difficult and time-consuming. For the other groups of rank at least 3, the specific Garside monoids chosen for these groups have been specified in [5], table 7.…”

Section: Introductionmentioning

confidence: 99%

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Homology computations for complex braid groups II

Marin

2017

Journal of Algebra

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Abstract. We complete the computation of the integral homology of the generalized braid group B associated to an arbitrary irreducible complex reflection group W of exceptional type. In order to do this we explicitely computed the recursively-defined differential of a resolution of Z as a ZB-module, using parallel computing. We also deduce from this general computation the rational homology of the Milnor fiber of the singularity attached to most of these reflection groups.

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“…By several arguments, recalled in [5], one can reduce the problem to a fewer number of groups. In particular, the homology of groups of small rank can be easily computed.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Homology computations for complex braid groups II

Marin

2017

Journal of Algebra

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“…This allows us to compute the center of B (k) (e, e, n). Finally, using the Garside structure of B (k) (e, e, n), we compute its first and second integral homology groups using the Dehornoy-Lafont complex [11] and the method used in [6].…”

Section: New Garside Structuresmentioning

confidence: 99%

Interval Garside structures for the complex braid groups 𝐵(𝑒,𝑒,𝑛)

Neaime

2019

Trans. Amer. Math. Soc.

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We define geodesic normal forms for the general series of complex reflection groups G(e, e, n). This requires the elaboration of a combinatorial technique in order to explicitly determine minimal word representatives of the elements of G(e, e, n) over the generating set of the presentation of Corran-Picantin. Using these geodesic normal forms, we construct intervals in G(e, e, n) that are lattices. This gives rise to interval Garside groups. We determine which of these groups are isomorphic to the complex braid group B(e, e, n) and get a complete classification. For the other Garside groups that appear in our construction, we provide some of their properties and compute their second integral homology groups in order to understand these new structures.A complex reflection is a linear transformation of finite order, which fixes a hyperplane pointwise. Let W be a finite subgroup of GL n (C) with n ≥ 1 and R be the set of complex reflections of W . We say that W is a complex reflection group if W is generated by R. It is well known that every complex reflection group is a direct product of irreducible ones. These irreducible complex reflection groups have been classified by Shephard and Todd [23] in 1954. The classification consists of the following cases:• The infinite series G(de, e, n) depending on three positive integer parameters,• 34 exceptional groups.As we are interested in the groups of the infinite series, we provide the definition of the group G(de, e, n). For the definition of the 34 exceptional groups, see [23].Definition 1.1. The group G(de, e, n) is defined as the group of n × n monomial matrices (each row and column has a unique nonzero entry), where• all nonzero entries are de-th roots of unity and• the product of all the nonzero entries is a d-th root of unity.Broué, Malle and Rouquier [5] managed to associate a complex braid group B to each complex reflection group W . The definition of the complex braid group related to W is as follows. Let A := {Ker(s − 1) s.t. s ∈ R} be the hyperplane arrangement and X := C n \ A be the hyperplane complement. The complex reflection group W acts naturally on X. Let X/W be its space of orbits.Definition 1.2. The complex braid group associated with W is defined as the fundamental group:B := π 1 (X/W ).If W is equal to G(de, e, n), then we denote by B(de, e, n) the associated complex braid group. According to the results in [5], one can readily check that the complex braid groups B(de, e, n) are all isomorphic to B(2e, e, n) for d > 1. Hence the complex braid groups associated with the 3-parameter series G(de, e, n) arise from the two 2-parameter series G(e, e, n) and G(2e, e, n). This paper concerns the complex braid groups B(e, e, n) and their associated complex reflection groups G(e, e, n).Let W be a real reflection group, meaning that W is a subgroup of GL n (R). By [3] and [2], we recover in the previous definitions the notion of Coxeter and Artin-Tits groups that we recall now. Definition 1.3. Assume that W is a group and S be a subset of W . For s and t in S,...

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“…We collect here some of the results concerning the homology of G An and G Bn with constant coefficients and with coefficients in abelian local systems. We follow the notation used in [CM14].…”

Section: Homology Of Some Artin Groupsmentioning

confidence: 99%

“…If the action of G Bn on the module M is trivial the spectral sequence collapses at E 2 . This fact can be proved with a quite technical argument: in fact one can see that all the non-zero differentials of the spectral sequence are divided by a coefficient p1`tq, where´t correspond to the action of the first standard generator of the group G Bn on the module M. In particular, if the action is trivial, all the differentials of the spectral sequence are trivial (see [CM14] for a detailed analysis of this spectral sequence). Another more elementary argument is the following: the splitting H˚pC 1,n q » H˚pC n q ' H˚´1pC 1,n-1 q induces the decomposition (see also [Gor78])…”

Section: ])mentioning

confidence: 99%

Homology of the family of hyperelliptic curves

Callegaro

Salvetti

2019

Isr. J. Math.

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Homology of braid groups and Artin groups can be related to the study of spaces of curves. We completely calculate the integral homology of the family of smooth curves of genus g with one boundary component, that are double coverings of the disk ramified over n " 2g`1 points. The main part of such homology is described by the homology of the braid group with coefficients in a symplectic representation, namely the braid group Br n acts on the first homology group of a genus g surface via Dehn twists. Our computations shows that such groups have only 2-torsion. We also investigate stabilization properties and provide Poincaré series, both for unstable and stable homology. 15 6. No 4-torsion 19 7. Stabilization and computations 21 References 25 2 6 7

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Homology computations for complex braid groups

Cited by 17 publications

References 24 publications

Homology computations for complex braid groups II

Homology computations for complex braid groups II

Interval Garside structures for the complex braid groups 𝐵(𝑒,𝑒,𝑛)

Homology of the family of hyperelliptic curves

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