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The Varchenko determinant for oriented matroids

  • Winfried Hochstättler
  • Volkmar WelkerEmail author
Article
  • 23 Downloads

Abstract

We generalize the Varchenko matrix of a hyperplane arrangement to oriented matroids. We show that the celebrated determinant formula for the Varchenko matrix, first proved by Varchenko, generalizes to oriented matroids. It follows that the determinant only depends on the matroid underlying the oriented matroid and analogous formulas hold for closed supertopes in oriented matroids. We follow a proof strategy for the original Varchenko formula first suggested by Denham and Hanlon. Besides several technical lemmas this strategy also requires a topological result on supertopes which is of independent interest. We show that a supertope considered as a subposet of the tope poset has a contractible order complex.

Keywords

Varchenko matrix Hyperplane arrangement Oriented matroid Supertope 

Mathematics Subject Classification

52C40 52B35 05B35 

Notes

Acknowledgements

The authors thank the referee for providing suggestions that helped to improve the exposition. Moreover, we are grateful for pointing us to COMs as a possible direction for generalizations.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Fakultät für Mathematik und InformatikFernUniversität in HagenHagenGermany
  2. 2.Fachbereich Mathematik und InformatikPhilipps-Universität MarburgMarburgGermany

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