The freeness conjecture for Hecke algebras of complex reflection groups, and the case of the Hessian group

Marin, Ivan

doi:10.1016/j.jpaa.2013.08.009

Cited by 33 publications

(58 citation statements)

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“…By general arguments (see, for example, [16]) this has for consequence the following: Corollary 1.2. If F is a suitably large extension of the field of fractions of R k , then H k ⊗ R k F is isomorphic to the group algebra F W k .…”

Section: Introductionmentioning

confidence: 89%

“…If we exclude the obvious cases n = 2 and k = 2, which lead to the cyclic groups and to the symmetric groups respectively, there is only a finite number of such groups, which are irreducible complex reflection groups: these are the groups G 4 , G 8 and G 16 , for n = 3 and k = 3, 4, 5 and the groups G 25 , G 32 for n = 4, 5 and k = 3, as they are known in the ShephardTodd classification (see [18]). Therefore, if we restrict ourselves to the case of B 3 , we have the finite quotients W k , for 2 ≤ k ≤ 5, which are the groups S 3 , G 4 , G 8 and G 16 , respectively.…”

Section: Introductionmentioning

confidence: 99%

“…These representations will be noted in GAP3 as R [15] and R [16]. We have θ(ω 4,5 (z 0 )) = −θ(ω 4,3 (z 0 )), which means that M is of defect zero i.e.…”

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confidence: 99%

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Universal deformations of the finite quotients of the braid group on 3 strands

Chavli

2016

Journal of Algebra

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Abstract. We prove that the quotients of the group algebra of the braid group on 3 strands by a generic quartic and quintic relation respectively, have finite rank. This is a special case of a conjecture by Broué, Malle and Rouquier for the generic Hecke algebra of an arbitrary complex reflection group. Exploring the consequences of this case, we prove that we can determine completely the irreducible representations of this braid group of dimension at most 5, thus recovering a classification of Tuba and Wenzl in a more general framework.Acknowledgments. I would like to thank my supervisor Ivan Marin for his help and support during this research, and Maria Chlouveraki, for fruitful discussions on the last section of this paper. I would also like to thank the University Paris Diderot -Paris 7 for its financial support.

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Section: Introductionmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

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Universal deformations of the finite quotients of the braid group on 3 strands

Chavli

2016

Journal of Algebra

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“…The BMR freeness conjecture is now proved for all irreducible reflection groups but the ones of Shephard-Todd types G 17 , G 18 and G 19 (see [3,4,17,15,20,6,7,8]), therefore the above statement is actually almost unconditional, and reduces the proof of conjecture 5.10 in [18] to the original BMR freeness conjecture.…”

Section: Introductionmentioning

confidence: 83%

Lattice extensions of Hecke algebras

Marin

2018

Journal of Algebra

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We investigate the extensions of the Hecke algebras of finite (complex) reflection groups by lattices of reflection subgroups that we introduced, for some of them, in our previous work on the Yokonuma-Hecke algebras and their connections with Artin groups. When the Hecke algebra is attached to the symmetric group, and the lattice contains all reflection subgroups, then these algebras are the diagram algebras of braids and ties of Aicardi and Juyumaya. We prove a stucture theorem for these algebras, generalizing a result of Espinoza and Ryom-Hansen from the case of the symmetric group to the general case. We prove that these algebras are symmetric algebras at least when W is a Coxeter group, and in general under the trace conjecture of Broué, Malle and Michel.

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“…Note that, thanks to the following result, which can be found in [BMR,Proof of Theorem 4.24] (for another detailed proof, one may also see [Mar3,Proposition 2.4]), in order to prove the BMR freeness conjecture, it is enough to find a spanning set of H(W ) over R(W ) consisting of |W | elements.…”

Section: Conjecturesmentioning

confidence: 99%