Recursively free reflection arrangements

Mücksch, Paul

doi:10.1016/j.jalgebra.2016.10.041

Cited by 5 publications

(3 citation statements)

References 11 publications

(10 reference statements)

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“…IF DF follows by Theorem 2.16. Moreover, by Lemma 3.13 in[8], we know that A(G 31 ) ∈ DF \ AF. Here G 31 is one of the finite unitary reflection groups classified and labeled by Shephard and Todd in[13].…”

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confidence: 90%

Addition-deletion theorem for free hyperplane arrangements and combinatorics

Abe¹

2018

Preprint

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In the theory of hyperplane arrangements, the most important and difficult problem is the combinatorial dependency of several properties. In this atricle, we prove that Terao's celebrated addition-deletion theorem for free arrangements is combinatorial, i.e., whether you can apply it depends only on the intersection lattice of arrangements. The proof is based on a classical technique. Since some parts are already completed recently, we prove the rest part, i.e., the combinatoriality of the addition theorem. As a corollary, we can define a new class of free arrangements called the additionally free arrangement of hyperplanes, which can be constructed from the empty arrangement by using only the addition theorem. Then we can show that Terao's conjecture is true in this class. As an application, we can show that every ideal-Shi arrangement is additionally free, implying that their freeness is combinatorial.

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confidence: 90%

Addition-deletion theorem for free hyperplane arrangements and combinatorics

Abe¹

2018

Preprint

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“…5 It is already know that such a matroid exists, namely the rank 3 reflection arrangement A(G 24 ) (with 21 hyperplanes) of the exceptional complex reflection group W = G 24 is recursively free [Mü17] but not inductively free [HR15]. Hence, an addition of A(G 24 ) is easily seen to be divisionally free but not inductively free.…”

Section: Introductionmentioning

confidence: 99%

On the generation of rank 3 simple matroids with an application to Terao's freeness conjecture

Barakat,

Behrends,

Jefferson

et al. 2019

Preprint

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In this paper we describe a parallel algorithm for generating all non-isomorphic rank 3 simple matroids with a given multiplicity vector. We apply our implementation in the HPC version of GAP to generate all rank 3 simple matroids with at most 14 atoms and a splitting characteristic polynomial. We have stored the resulting matroids alongside with various useful invariants in a publicly available, ArangoDB-powered database. As a byproduct we show that the smallest divisionally free rank 3 arrangement which is not inductively free has 14 hyperplanes and exists in all characteristics distinct from 2 and 5. Another database query proves that Terao's freeness conjecture is true for rank 3 arrangements with 14 hyperplanes in any characteristic.

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“…For applications of Theorem 1.1 for RF and Theorem 1.2 in the context of the classification of recursively free reflection arrangements, see [Mü15].…”

Section: Introductionmentioning

confidence: 99%

Inductive and Recursive Freeness of Localizations of Multiarrangements

Hoge

Röhrle

Schauenburg

2017

Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory

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The class of free multiarrangements is known to be closed under taking localizations. We extend this result to the stronger notions of inductive and recursive freeness.As an application, we prove that recursively free multiarrangements are compatible with the product construction for multiarrangements. In addition, we show how our results can be used to derive that some canonical classes of free multiarrangements are not inductively free.

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Recursively free reflection arrangements

Cited by 5 publications

References 11 publications

Addition-deletion theorem for free hyperplane arrangements and combinatorics

Addition-deletion theorem for free hyperplane arrangements and combinatorics

On the generation of rank 3 simple matroids with an application to Terao's freeness conjecture

Inductive and Recursive Freeness of Localizations of Multiarrangements

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