On the homology of the noncrossing partition lattice and the Milnor fibre

Zhang, Yang

doi:10.1016/j.aim.2023.108897

Cited by 1 publication

(7 citation statements)

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“…We next give what later will turn out to be a combinatorial definition of the noncrossing algebra above, as introduced in [Zha23].…”

Section: The Algebra Bmentioning

confidence: 99%

“…Proposition 2.8. [Zha23,Proposition 3.4] Let w ∈ L k with k ≥ 1. Then for any t ∈ Rex T (w), we have…”

Section: The Algebra Bmentioning

confidence: 99%

“…Evidently W acts on both M and F , so that we may speak of the orbit spaces M/W and F/W . [Zha23] the following chain complexes which compute the integral homology of the hyperplane complement and of the Milnor fibre. We use the algebra A instead of B in the original complexes.…”

Section: Complexes For the Milnor Fibre And Hyperplane Complementmentioning

confidence: 99%

“…This work is an outgrowth of [Zha23], whose notation we follow, by and large. In particular, W is a finite Coxeter group, M is its corresponding complexified hyperplane complement and F is the corresponding (non-reduced) Milnor fibre, as defined in [DL16, Def.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, W is a finite Coxeter group, M is its corresponding complexified hyperplane complement and F is the corresponding (non-reduced) Milnor fibre, as defined in [DL16, Def. 1] or [Zha23]. Our objective is to construct tractable chain complexes which compute the homology of M (which is known for all W ) and that of F (which is poorly understood, even in the case W = Sym n ).…”

Section: Introductionmentioning

confidence: 99%

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Milnor fibre homology complexes

Lehrer,

Zhang

2024

Pure and Applied Mathematics Quarterly

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Let W be a finite Coxeter group. We give an algebraic presentation of what we refer to as "the non-crossing algebra", which is associated to the hyperplane complement of W and to the cohomology of its Milnor fibre. This is used to produce simpler and more general chain (and cochain) complexes which compute the integral homology and cohomology groups of the Milnor fibre F of W . In the process we define a new, larger algebra A, which seems to be "dual" to the Fomin-Kirillov algebra, and in low ranks is linearly isomorphic to it. There is also a mysterious connection between A and the Orlik-Solomon algebra, in analogy with the fact that the Fomin-Kirillov algebra contains the coinvariant algebra of W . This analysis is applied to compute the multiplicities ρ, H k (F, C) W and ρ, H k (M, C) W , where M and F are respectively the hyperplane complement and Milnor fibre associated to W and ρ is a representation of W .

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“…We next give what later will turn out to be a combinatorial definition of the noncrossing algebra above, as introduced in [Zha23].…”

Section: The Algebra Bmentioning

confidence: 99%

“…Proposition 2.8. [Zha23,Proposition 3.4] Let w ∈ L k with k ≥ 1. Then for any t ∈ Rex T (w), we have…”

Section: The Algebra Bmentioning

confidence: 99%

Section: Complexes For the Milnor Fibre And Hyperplane Complementmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning