### Abstract

Let W be a finite Weyl group and A be the corresponding Weyl arrangement. A deformation of A is an affine arrangement which is obtained by adding to each hyperplane H∈A several parallel translations of H by the positive root (and its integer multiples) perpendicular to H. We say that a deformation is W-equivariant if the number of parallel hyperplanes of each hyperplane H∈A depends only on the W-orbit of H. We prove that the conings of the W-equivariant deformations are free arrangements under a Shi-Catalan condition and give a formula for the number of chambers. This generalizes Yoshinaga's theorem conjectured by Edelman-Reiner.

Original language | English |
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Pages (from-to) | 1191-1198 |

Number of pages | 8 |

Journal | European Journal of Combinatorics |

Volume | 32 |

Issue number | 8 |

DOIs | |

Publication status | Published - Nov 1 2011 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics

### Cite this

*European Journal of Combinatorics*,

*32*(8), 1191-1198. https://doi.org/10.1016/j.ejc.2011.06.005

**The freeness of Shi-Catalan arrangements.** / Abe, Takuro; Terao, Hiroaki.

Research output: Contribution to journal › Article

*European Journal of Combinatorics*, vol. 32, no. 8, pp. 1191-1198. https://doi.org/10.1016/j.ejc.2011.06.005

}

TY - JOUR

T1 - The freeness of Shi-Catalan arrangements

AU - Abe, Takuro

AU - Terao, Hiroaki

PY - 2011/11/1

Y1 - 2011/11/1

N2 - Let W be a finite Weyl group and A be the corresponding Weyl arrangement. A deformation of A is an affine arrangement which is obtained by adding to each hyperplane H∈A several parallel translations of H by the positive root (and its integer multiples) perpendicular to H. We say that a deformation is W-equivariant if the number of parallel hyperplanes of each hyperplane H∈A depends only on the W-orbit of H. We prove that the conings of the W-equivariant deformations are free arrangements under a Shi-Catalan condition and give a formula for the number of chambers. This generalizes Yoshinaga's theorem conjectured by Edelman-Reiner.

AB - Let W be a finite Weyl group and A be the corresponding Weyl arrangement. A deformation of A is an affine arrangement which is obtained by adding to each hyperplane H∈A several parallel translations of H by the positive root (and its integer multiples) perpendicular to H. We say that a deformation is W-equivariant if the number of parallel hyperplanes of each hyperplane H∈A depends only on the W-orbit of H. We prove that the conings of the W-equivariant deformations are free arrangements under a Shi-Catalan condition and give a formula for the number of chambers. This generalizes Yoshinaga's theorem conjectured by Edelman-Reiner.

UR - http://www.scopus.com/inward/record.url?scp=81155157637&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=81155157637&partnerID=8YFLogxK

U2 - 10.1016/j.ejc.2011.06.005

DO - 10.1016/j.ejc.2011.06.005

M3 - Article

VL - 32

SP - 1191

EP - 1198

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

SN - 0195-6698

IS - 8

ER -