Braid groups of imprimitive complex reflection groups

Corran, Ruth; Lee, Eon-Kyung; Lee, Sang-Jin

doi:10.1016/j.jalgebra.2015.01.004

Cited by 13 publications

(19 citation statements)

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“…The last fact can also be proved group theoretically ( [15]). The Artin groups corresponding to the finite complex reflection groups G(de, e, r) is torsion free, since A G(de,e,r) is a subgroup of A Br ([ [7], Proposition 4.1]).…”

Section: Pa=pure Artin Group Of Typementioning

confidence: 99%

“…For the proof of the theorem in the G(de, e, r)-case we refer to Remark 2.1. In ([ [7], Proposition 4.1]) it is shown that A G(de,e,r) is a subgroup of A Br . (In [7], these are called braid groups associated to the reflection groups).…”

Section: Proofs Of the Theoremsmentioning

confidence: 99%

“…In ([ [7], Proposition 4.1]) it is shown that A G(de,e,r) is a subgroup of A Br . (In [7], these are called braid groups associated to the reflection groups). Therefore, using Condition (2) and the B n case of Theorem 3.1, A G(de,e,r) ∈ C.…”

Section: Proofs Of the Theoremsmentioning

confidence: 99%

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A certain structure of Artin groups and the isomorphism conjecture

Roushon

2018

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We observe an inductive structure in a large class of Artin groups and exploit this information to deduce the Farrell-Jones isomorphism conjecture for several classes of Artin groups of finite real, complex and affine types.As an application, in Theorem 3.4, we extend our earlier computation of the surgery groups of the pure braid groups in [29], to the finite type pure Artin groups appearing in Theorem 1.1.Corollary 1.1. For any subgroup of Γ, where Γ is as in Theorem 1.1 the two assembly maps A K and A L are isomorphisms.Below we mention a well known consequence of the isomorphism of the K-theory assembly map. See [[12], §1.6.1].Corollary 1.2. The Whitehead group W h(−), the reduced projective class group K0 (−) and the negative K-groups K −i (−), for i ≥ 1, vanish for any subgroup of the groups considered in Theorem 1.1.The paper is organized as follows. In Section 2 we recall some background on Artin groups and in Section 3 we state our results which are used to prove the main theorem. We describe orbifold braid groups and some work from [1] in Section 4, which is the key input to this work. Section 5 contains the proofs of the results of Section 3. The computation of the surgery groups of pure Artin groups is given in Section 6. Finally, in Section 7 we recall the statement of the isomorphism conjecture and all the basic facts needed to prove Theorem 3.3.We end the Introduction with the following remark.

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Section: Pa=pure Artin Group Of Typementioning

confidence: 99%

Section: Proofs Of the Theoremsmentioning

confidence: 99%

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A certain structure of Artin groups and the isomorphism conjecture

Roushon

2018

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“…This section sets the stage for our later work by establishing a set of geodesic normal forms for the complex reflection groups G(de, e, n), using the generating sets introduced by Corran-Picantin [10] and Corran-Lee-Lee [9] for G(e, e, n) and G(de, e, n) for d > 1, respectively. The case of G(e, e, n) has been already done in our previous work (see Section 3 of [17]).…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

“…In order to establish these bases, our attention is firstly shifted to the complex reflection groups G(de, e, n). We construct geodesic normal forms for these groups by using the presentations of Corran-Picantin [10] and Corran-Lee-Lee [9] of G(e, e, n) and G(de, e, n) for d > 1, respectively. The geodesic normal forms are very natural and easy to describe.…”

Section: Introductionmentioning

confidence: 99%

Geodesic Normal Forms and Hecke Algebras for the Complex Reflection Groups G(de,e,n)

Neaime¹

2018

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We establish geodesic normal forms for the general series of complex reflection groups G(de, e, n) by using the presentations of Corran-Picantin and Corran-Lee-Lee of G(e, e, n) and G(de, e, n) for d > 1, respectively. This requires the elaboration of a combinatorial technique in order to explicitly determine minimal word representatives of the elements of G(de, e, n). Using these geodesic normal forms, we construct natural bases for the Hecke algebras associated with the complex reflection groups G(e, e, n) and G(d, 1, n). As an application, we obtain a new proof of the BMR freeness conjecture for these groups.Note that it is enough to choose one such relation per conjugacy class of distinguished reflections, as all the corresponding braided reflections are conjugates in B (see [6]).The BMR freeness conjecture proposed by Broué, Malle and Rouquier [6] in 1998 states that the Hecke algebra H(W ) attached to W is a free R-module of rank equal to the order of W . After two decades, the BMR freeness conjecture is proven through the results of a number of authors. Thus, we have the following theorem.Theorem 2.3. The Hecke algebra H(W ) is a free R-module of rank |W |.The BMR freeness conjecture can be easily reduced to the case where W is irreducible. It is true for the (Iwahori-)Hecke algebra attached to any finite Coxeter group (see Lemma 4.4.3 of [11]). Ariki and Koike [2] proved it for the case of G(d, 1, n). Note that a basis for the Hecke algebra associated with G(d, 1, n) is also given in [4]. Ariki defined in [1] a Hecke algebra for G(de, e, n) by a presentation with generators and relations. He also proved that it is a free module of rank |G(de, e, n)|. The Hecke algebra defined by Ariki is isomorphic to the Hecke algebra defined by Broué, Malle, and Rouquier in [6] for G(de, e, n). The details why this is true can be found in Appendix A.2 of [18]. Hence one gets a proof of Theorem 2.3 for the general series of complex reflection groups.Concerning the exceptional complex reflection groups, Marin proved the conjecture for G 4 , G 25 , G 26 , and G 32 in [12] and [13]. Marin and Pfeiffer proved it for G 12 , G 22 , G 24 , G 27 , G 29 , G 31 , G 33 , and G 34 in [15]. In her PhD thesis and in the article that followed (see [7] and [8]), Chavli proved the validity of this conjecture for G 5 , G 6 , • • • , G 16 . Recently, Marin proved the conjecture for G 20 and G 21 (see [14]) and finally Tsushioka for G 17 , G 18 and G 19 (see [20]). Hence we obtain a proof of Theorem 2.3 for all the cases of irreducible complex reflection groups.

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Configuration Lie groupoids and orbifold braid groups

Roushon¹

2021

Bulletin des Sciences Mathématiques

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Braid groups of imprimitive complex reflection groups

Cited by 13 publications

References 46 publications

A certain structure of Artin groups and the isomorphism conjecture

A certain structure of Artin groups and the isomorphism conjecture

Geodesic Normal Forms and Hecke Algebras for the Complex Reflection Groups G(de,e,n)

Configuration Lie groupoids and orbifold braid groups

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