Israel Journal of Mathematics

, Volume 150, Issue 1, pp 229–240 | Cite as

Cyclotomic and simplicial matroids

  • Jeremy L. Martin
  • Victor Reiner


We show that two naturally occurring matroids representable over ℚ are equal: thecyclotomic matroid µn represented by then th roots of unity 1, ζ, ζ2, …, ζn-1 inside the cyclotomic extension ℚ(ζ), and a direct sum of copies of a certain simplicial matroid, considered originally by Bolker in the context of transportation polytopes. A result of Adin leads to an upper bound for the number of ℚ-bases for ℚ(ζ) among then th roots of unity, which is tight if and only ifn has at most two odd prime factors. In addition, we study the Tutte polynomial of µn in the case thatn has two prime factors.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    R. M. Adin,Counting colorful multi-dimensional trees, Combinatorica12 (1992), 247–260.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    F. Ardila,Enumerative and algebraic aspects of matroids and hyperplane arrangements, Ph.D. thesis, MIT, 2003.Google Scholar
  3. [3]
    F. Ardila,A tropical morphism related to the hyperplane arrangement of the complete bipartite graph, Preprint, arXiv: math. CO/0404287, 2004.Google Scholar
  4. [4]
    L. W. Beineke and R. E. Pippert,Properties and characterizations of k-trees, Mathematika18 (1971), 141–151.MathSciNetMATHGoogle Scholar
  5. [5]
    E. D. Bolker,Simplicial geometry and transportation polytopes, Transactions of the American Mathematical Society217 (1976), 121–142.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    H. H. Crapo and G.-C. Rota,On the Foundations of Combinatorial Theory: Combinatorial Geometries, Preliminary edition, The M.I.T. Press, Cambridge, Mass.-London, 1970.MATHGoogle Scholar
  7. [7]
    S. Faridi,The facet ideal of a simplicial complex, Manuscripta Mathematica109 (2002), 159–174.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    F. Harary and E. M. Palmer,On acyclic simplicial complexes, Mathematika15 (1968), 115–122.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    K. Johnsen,Lineare Abhängigkeiten von Einheitswurzeln, Elemente der Mathematik40 (1985), 57–59.MATHMathSciNetGoogle Scholar
  10. [10]
    G. Kalai,Enumeration of ℚ-acyclic simplicial complexes, Israel Journal of Mathematics45 (1983), 337–351.MATHMathSciNetGoogle Scholar
  11. [11]
    B. Lindström,Matroids on the bases of simple matroids, European Journal of Combinatorics2 (1981), 61–63.MATHMathSciNetGoogle Scholar
  12. [12]
    J. R. Munkres,Elements of Algebraic Topology, Addison-Wesley, Menlo Park, CA, 1984.MATHGoogle Scholar
  13. [13]
    J. G. Oxley,matroid Theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1992.MATHGoogle Scholar
  14. [14]
    R. P. Stanley,Enumerative Combinatorics, Vol. 2, Cambridge University Press, Cambridge, 1999.Google Scholar
  15. [15]
    D. J. A. Welsh,Matroid Theory, London Mathematical Society Monographs8, Academic Press, London-New York, 1976.MATHGoogle Scholar
  16. [16]
    N. White,Theory of matroids, inEncyclopedia of Mathematics and its Applications 26, Cambridge University Press, Cambridge, 1986.Google Scholar
  17. [17]
    N. White,Combinatorial geometries, inEncyclopedia of Mathematics and its Applications 29, Cambridge University Press, Cambridge, 1987.Google Scholar
  18. [18]
    N. White,Matroid applications, inEncyclopedia of Mathematics and its Applications 40, Cambridge University Press, Cambridge, 1992.Google Scholar

Copyright information

© Hebrew University 2005

Authors and Affiliations

  • Jeremy L. Martin
    • 1
  • Victor Reiner
    • 1
  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA

Personalised recommendations