Tohoku Mathematical Journal

Reflection arrangements are hereditarily free

Torsten Hoge and Gerhard Röhrle

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Abstract

Suppose that $W$ is a finite, unitary, reflection group acting on the complex vector space $V$. Let ${\mathcal A} = {\mathcal A}(W)$ be the associated hyperplane arrangement of $W$. Terao has shown that each such reflection arrangement ${\mathcal A}$ is free. Let $L({\mathcal A})$ be the intersection lattice of ${\mathcal A}$. For a subspace $X$ in $L({\mathcal A})$ we have the restricted arrangement ${\mathcal A}^X$ in $X$ by means of restricting hyperplanes from ${\mathcal A}$ to $X$. In 1992, Orlik and Terao conjectured that each such restriction is again free. In this note we settle the outstanding cases confirming the conjecture.

In 1992, Orlik and Terao also conjectured that every reflection arrangement is hereditarily inductively free. In contrast, this stronger conjecture is false however; we give two counterexamples.

Article information

Source
Tohoku Math. J. (2), Volume 65, Number 3 (2013), 313-319.

Dates
First available in Project Euclid: 12 September 2013

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1378991017

Digital Object Identifier
doi:10.2748/tmj/1378991017

Mathematical Reviews number (MathSciNet)
MR3102536

Zentralblatt MATH identifier
1287.51006

Subjects
Primary: 20F55: Reflection and Coxeter groups [See also 22E40, 51F15]
Secondary: 52B30 52C35: Arrangements of points, flats, hyperplanes [See also 32S22] 14N20: Configurations and arrangements of linear subspaces 13N15: Derivations

Keywords
Complex reflection groups Freeness of restrictions of reflection arrangements

Citation

Hoge, Torsten; Röhrle, Gerhard. Reflection arrangements are hereditarily free. Tohoku Math. J. (2) 65 (2013), no. 3, 313--319. doi:10.2748/tmj/1378991017. https://projecteuclid.org/euclid.tmj/1378991017


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References

  • M. Barakat and M. Cuntz, Coxeter and crystallographic arrangements are inductively free, Adv. Math 229 (2012), 691–709.
  • N. Bourbaki, Éléments de mathématique. Groupes et algèbres de Lie. Chapitre IV-VI, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968.
  • J. M. Douglass, The adjoint representation of a reductive group and hyperplane arrangements, Represent. Theory 3 (1999), 444–456.
  • M. Geck, G. Hiß, F. Lübeck, G. Malle and G. Pfeiffer, CHEVIE –- A system for computing and processing generic character tables, Appl. Algebra Engrg. Comm. Comput. 7 (1996), 175–210.
  • G.-M. Greuel, G. Pfister and H. Schönemann, Singular 3-1-1, A Computer Algebra System for Polynomial Computations, Centre for Computer Algebra, University of Kaiserslautern, 2009.
  • T. Hoge and G. Röhrle, On inductively free reflection arrangements, J. Reine Angew. Math., in press; DOI: 10.1515/crelle-2013-0022.
  • P. Orlik and H. Terao, Arrangements of hyperplanes, Springer-Verlag, Berlin, 1992.
  • P. Orlik and H. Terao, Coxeter arrangements are hereditarily free, Tôhoku Math. J. 45 (1993), 369–383.
  • K. Saito, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), no. 2, 265–291.
  • M. Schönert et al., GAP - Groups, Algorithms, and Programming – version 3 release 4, 1997.
  • M. Schulze, Freeness and multirestriction of hyperplane arrangements, Compositio Math. 148 (2012), 799–806.
  • H. Terao, Arrangements of hyperplanes and their freeness I, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), 293–320.