The Varchenko determinant for oriented matroids

  • Winfried Hochstättler
  • Volkmar WelkerEmail author


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.


Varchenko matrix Hyperplane arrangement Oriented matroid Supertope 

Mathematics Subject Classification

52C40 52B35 05B35 



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.


  1. 1.
    Aguiar, M., Mahajan, S.: Topics in Hyperplane Arrangements. Mathematical Surveys and Monographs, vol. 226. American Mathematical Society, Providence (2017)zbMATHGoogle Scholar
  2. 2.
    Bandelt, H.J., Chepoi, V., Knauer, K.: COMs: complexes of oriented matroids. J. Comb. Theory Ser. A 156, 195–237 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Björner, A.: Topological methods. In: Graham, R.L., Grötschel, M., Lovász, L. (eds.) Handbook of Combinatorics, vol. 2, pp. 1819–1872. Elsevier Science B.V., Amsterdam (1995)Google Scholar
  4. 4.
    Björner, A., Las Vergnas, M., Sturmfels, B., White, N., Ziegler, G.M.: Oriented Matroids. Encyclopedia of Mathematics and Its Applications, vol. 46, 2nd edn. Cambridge University Press, Cambridge (1999)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bryławski, T., Varchenko, A.: The determinant formula for a matroid bilinear form. Adv. Math. 129, 1–24 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Denham, G., Hanlon, P.: Some algebraic properties of the Schechtman–Varchenko bilinear forms. In: Billera, L.J., Björner, A., Greene, C., Simion, R.E., Stanley, R.P. (eds.) New Perspectives in Algebraic Combinatorics (Berkeley, CA, 199697). Mathematical Sciences Research Institute Publications, vol. 38, pp. 149–176. Cambridge University Press, Cambridge (1999)Google Scholar
  7. 7.
    Gente, R.: The Varchenko matrix for cones. PhD-Thesis, Philipps-Universität Marburg (2013)Google Scholar
  8. 8.
    Hanlon, P., Stanley, R.P.: A q-deformation of a trivial symmetric group action. Trans. Am. Math. Soc. 350, 4445–4459 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Pfeiffer, G., Randriamaro, H.: The Varchenko determinant of a Coxeter arrangement. J. Group Theory 21, 651–665 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Schechtman, V.V., Varchenko, A.N.: Arrangements of hyperplanes and Lie algebra homology. Invent. Math. 106, 139–194 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Schechtman, V.V., Varchenko, A.N.: Quantum groups and homology of local systems. In: Fujiki, A., Kato, K., Katsura, T., Kawamata, Y., Miyaoka, Y. (eds.) Algebraic Geometry and Analytic Geometry (Tokyo, 1990), pp. 182–197. Springer, Tokyo (1991)Google Scholar
  12. 12.
    Stanley, R.P.: Enumerative Combinatorics. Volume 1. Cambridge Studies in Advanced Mathematics, vol. 49, 2nd edn. Cambridge University Press, Cambridge (2012)zbMATHGoogle Scholar
  13. 13.
    Varchenko, A.N.: Bilinear form of real configuration of hyperplanes. Adv. Math. 97, 110–144 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Vembar, N.: An oriented matroid determinant. Master Thesis, University of North Carolina, Chapel Hill (2001)Google Scholar
  15. 15.
    Vembar, N.: Oriented matroid integer chains. PhD-Thesis, University of North Carolina, Chapel Hill (2003)Google Scholar
  16. 16.
    Zagier, D.: Realizability of a model in infinite statistics. Commun. Math. Phys. 147, 199–210 (1992)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

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

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