scholarly journals The double Coxeter arrangement

1998 ◽  
Vol 73 (2) ◽  
pp. 237-258 ◽  
Author(s):  
L. Solomon ◽  
H. Terao
Keyword(s):  
Coxeter Arrangement

scholarly journals k-Parabolic Subspace Arrangements

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Christopher Severs ◽  
Jacob White
Keyword(s):  
Fundamental Group ◽  
Reflection Group ◽  
Artin Group ◽  
Subspace Arrangements ◽  
Coxeter Arrangement ◽  
Algebraic Description ◽  
Finite Reflection Group ◽  
International Audience

International audience In this paper, we study k-parabolic arrangements, a generalization of the k-equal arrangement for any finite real reflection group. When k=2, these arrangements correspond to the well-studied Coxeter arrangements. Brieskorn (1971) showed that the fundamental group of the complement of the type W Coxeter arrangement (over $\mathbb{C}$) is isomorphic to the pure Artin group of type W. Khovanov (1996) gave an algebraic description for the fundamental group of the complement of the 3-equal arrangement (over $\mathbb{R}$). We generalize Khovanov's result to obtain an algebraic description of the fundamental group of the complement of the 3-parabolic arrangement for arbitrary finite reflection group. Our description is a real analogue to Brieskorn's description. Nous généralisons les arrangements k-égaux à tous les groupes de réflexions finis réels. Les arrangements ainsi obtenus sont dits k-paraboliques. Dans le cas où k = 2 nous retrouvons les arrangements de Coxeter qui sont bien connus. En 1971, Brieskorn démontra que le groupe fondamental associé au complément (complexe) de l'arrangement de Coxeter de type W est en fait isomorphe au groupe pure d'Artin de type W . En 1996, Khovanov donne une description algébrique du groupe fondamental du complément (réel) de l'arrangement 3-égaux. Nous généralisons le résultat de Khovanov et obtenons une description algébrique du groupe fondamental de l'espace complément d'un arrangement k-parabolique pour tous les groupes de réflexions finis et réels. Il se trouve que notre description est l'analogue réel de la description de Brieskorn.

scholarly journals Equivariant Log Concavity and Representation Stability

2021 ◽  
Author(s):  
Jacob P Matherne ◽  
Dane Miyata ◽  
Nicholas Proudfoot ◽  
Eric Ramos
Keyword(s):  
Lie Algebra ◽  
Hyperplane Arrangements ◽  
Oriented Matroids ◽  
Computer Assisted ◽  
Representation Stability ◽  
Low Degree ◽  
Coxeter Arrangement

Abstract We expand upon the notion of equivariant log concavity and make equivariant log concavity conjectures for Orlik–Solomon algebras of matroids, Cordovil algebras of oriented matroids, and Orlik–Terao algebras of hyperplane arrangements. In the case of the Coxeter arrangement for the Lie algebra $\mathfrak{s}\mathfrak{l}_n$, we exploit the theory of representation stability to give computer-assisted proofs of these conjectures in low degree.

scholarly journals The Varchenko determinant of a Coxeter arrangement

2018 ◽  
Vol 21 (4) ◽  
pp. 651-665 ◽  
Author(s):  
Götz Pfeiffer ◽  
Hery Randriamaro
Keyword(s):  
Explicit Formula ◽  
Coxeter Groups ◽  
Hyperplane Arrangements ◽  
Arrangement Of Hyperplanes ◽  
Coxeter Arrangement

AbstractThe Varchenko determinant is the determinant of a matrix defined from an arrangement of hyperplanes. Varchenko proved that this determinant has a beautiful factorization. It is, however, not possible to use this factorization to compute a Varchenko determinant from a certain level of complexity. Precisely at this point, we provide an explicit formula for this determinant for the hyperplane arrangements associated to the finite Coxeter groups. The intersections of hyperplanes with the chambers of such arrangements have nice properties which play a central role for the calculation of their associated determinants.

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