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Relation spaces of hyperplane arrangements and modules defined by graphs of fiber zonotopes

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Abstract

We study the exactness of certain combinatorially defined complexes which generalize the Orlik-Solomon algebra of a geometric lattice. The main results pertain to complex reflection arrangements and their restrictions. In particular, we consider the corresponding relation complexes and give a simple proof of the n-formality of these hyperplane arrangements. As an application, we are able to bound the Castelnouvo-Mumford regularity of certain modules over polynomial rings associated to Coxeter arrangements (real reflection arrangements) and their restrictions. The modules in question are defined using the relation complex of the Coxeter arrangement and fiber polytopes of the dual Coxeter zonotope. They generalize the algebra of piecewise polynomial functions on the original arrangement.

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Authors and Affiliations

  1. Institut für Mathematik, Freie Universität Berlin, Arnimallee 3, D-14195, Berlin, Germany

    Tobias Finis

  2. Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel

    Erez Lapid

  3. Department of Mathematics, Weizmann Institute of Science, Rehovot, 76100, Israel

    Erez Lapid

Authors
  1. Tobias Finis
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  2. Erez Lapid
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Correspondence to Tobias Finis.

Additional information

Authors partially sponsored by grant # 964-107.6/2007 from the German-Israeli Foundation for Scientific Research and Development. First-named author supported by DFG Heisenberg grant # FI 1795/1-1.

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Finis, T., Lapid, E. Relation spaces of hyperplane arrangements and modules defined by graphs of fiber zonotopes. Isr. J. Math. 201, 901–947 (2014). https://doi.org/10.1007/s11856-014-1052-9

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  • DOI: https://doi.org/10.1007/s11856-014-1052-9

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