manuscripta mathematica

, Volume 156, Issue 1–2, pp 187–213 | Cite as

A Grassmann algebra for matroids

  • Jeffrey GiansiracusaEmail author
  • Noah Giansiracusa


We introduce an idempotent analogue of the exterior algebra for which the theory of tropical linear spaces (and valuated matroids) can be seen in close analogy with the classical Grassmann algebra formalism for linear spaces. The top wedge power of a tropical linear space is its Plücker vector, which we view as a tensor, and a tropical linear space is recovered from its Plücker vector as the kernel of the corresponding wedge multiplication map. We prove that an arbitrary d-tensor satisfies the tropical Plücker relations (valuated exchange axiom) if and only if the dth wedge power of the kernel of wedge-multiplication is free of rank one. This provides a new cryptomorphism for valuated matroids, including ordinary matroids as a special case.

Mathematics Subject Classification

05B35 15A75 15A80 15A15 14T05 12K10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bergman, G.M.: The logarithmic limit-set of an algebraic variety. Trans. Am. Math. Soc. 157, 459–469 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Crapo, H., Schmitt, W.: The Whitney algebra of a matroid. J. Comb. Theory Ser. A 91(1–2), 215–263 (2000) (In memory of Gian-Carlo Rota)Google Scholar
  3. 3.
    Dress, A.W.M.: Duality theory for finite and infinite matroids with coefficients. Adv. Math. 59(2), 97–123 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dress, A.W.M., Wenzel, W.: Valuated matroids: a new look at the greedy algorithm. Appl. Math. Lett. 3(2), 33–35 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dress, A., Wenzel, W.: Grassmann–Plücker relations and matroids with coefficients. Adv. Math. 86(1), 68–110 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dress, A.W.M., Wenzel, W.: Valuated matroids. Adv. Math. 93(2), 214–250 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fink, A., Rincón, F.: Stiefel tropical linear spaces. J. Comb. Theory Ser. A 135, 291–331 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Frenk, B.: Tropical Varieties, Maps and Gossip. Ph.D. thesis, Eindhoven University of Technology (2013)Google Scholar
  9. 9.
    Feichtner, E.M., Sturmfels, B.: Matroid polytopes, nested sets and Bergman fans. Port. Math. (N.S.) 62(4), 437–468 (2005)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Giansiracusa, J., Giansiracusa, N.: Equations of tropical varieties. Duke Math. J. 165(18), 3379–3433 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Golan, J.S.: Semirings and Affine Equations Over Them: Theory and Applications, Mathematics and Its Applications, vol. 556. Kluwer Academic Publishers Group, Dordrecht (2003)CrossRefzbMATHGoogle Scholar
  12. 12.
    Grassmann, H.: Gesammelte mathematische und physikalische Werke. Ersten Bandes, erster Theil: Die Ausdehnungslehre von 1844 und die geometrische Analyse, Unter der Mitwirkung von Eduard Study. Herausgegeben von Friedrich Engel. Chelsea Publishing Co., New York (1969)Google Scholar
  13. 13.
    Katz, E.: Matroid theory for algebraic geometers. In: Baker, M., Payne, S. (eds.) Nonarchimedean and Tropical Geometry Simons Symposia. Springer, Berlin (2016)Google Scholar
  14. 14.
    Mikhalkin, G.: Tropical geometry and its applications. In: International Congress of Mathematicians, vol. II, pp 827–852. European Mathematical Society, Zurich (2006)Google Scholar
  15. 15.
    Minc, H.: Permanents, Encyclopedia of Mathematics and its Applications, vol. 6. Addison-Wesley Publishing Co., Reading, Mass. (1978)Google Scholar
  16. 16.
    Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry, Graduate Studies in Mathematics, vol. 161. American Mathematical Society, Providence, RI (2015)CrossRefzbMATHGoogle Scholar
  17. 17.
    Murota, K., Tamura, A.: On circuit valuation of matroids. Adv. Appl. Math. 26(3), 192–225 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Murota, K.: Matroid valuation on independent sets. J. Comb. Theory Ser. B 69(1), 59–78 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Orlik, P., Solomon, L.: Combinatorics and topology of complements of hyperplanes. Invent. Math. 56(2), 167–189 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Oxley, J.: Matroid Theory. Oxford Graduate Texts in Mathematics, vol. 21, 2nd edn. Oxford University Press, Oxford (2011)CrossRefzbMATHGoogle Scholar
  21. 21.
    Payne, S.: Analytification is the limit of all tropicalizations. Math. Res. Lett. 16(3), 543–556 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Richter-Gebert, J., Sturmfels, B., Theobald, T.: First steps in tropical geometry, idempotent mathematics and mathematical physics. Contemp. Math., vol. 377, pp. 289–317. American Mathematical Society, Providence, RI (2005)Google Scholar
  23. 23.
    Speyer, D.: Tropical linear spaces. SIAM J. Discrete Math. 22(4), 1527–1558 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Speyer, D., Sturmfels, B.: The tropical Grassmannian. Adv. Geom. 4(3), 389–411 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    White, N. (ed.): Theory of Matroids. Encyclopedia of Mathematics and its Applications, vol. 26. Cambridge University Press, Cambridge (1986)Google Scholar
  26. 26.
    White, N. (ed.): Matroid Applications. Encyclopedia of Mathematics and its Applications, vol. 40. Cambridge University Press, Cambridge (1992)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.College of ScienceSwansea UniversitySwanseaUK
  2. 2.Department of Mathematics and StatisticsSwarthmore CollegeSwartmoreUSA

Personalised recommendations