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Annals of Combinatorics

, 15:37 | Cite as

A ‘Non-Additive’ Characterization of \({\wp}\)-Adic Norms

  • A. DressEmail author
  • J. Kåhrström
  • V. Moulton
Article
  • 59 Downloads

Abstract

For F a \({\wp}\)-adic field together with a \({\wp}\)-adic valuation, we present a new characterization for a map \({p: F^{n} \rightarrow {\mathbb R}\cup\{-\infty}\}\) to be a \({\wp}\)-adic norm on the vector space F n . This characterization was motivated by the concept of tight maps — maps that naturally arise within the theory of valuated matroids and tight spans. As an immediate consequence, we show that the two descriptions of the affine building of SL n (F) in terms of (i) \({\wp}\)-adic norms given by Bruhat and Tits and (ii) tight maps given by Terhalle essentially coincide. The result suggests that similar characterizations of affine buildings of other classical groups should exist, and that the theory of affine buildings may turn out as a particular case of a yet to be developed geometric theory of valuated (and δ-valuated) matroids and their tight spans providing simply-connected G-spaces for large classes of appropriately specified groups G that could serve as a basis for an affine variant of Gromov’s theory.

Mathematics Subject Classification

20E42 

Keywords

buildings affine buildings \({\wp}\)-adic norms tropical geometry of Grassmann-Plücker varieties matroids valuated matroids tight span 

References

  1. 1.
    Bachem A., Dress A., Wenzel W.: Five variations on a theme by Gyula Farkas. Adv. App. Math. 13(2), 160–185 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bogart T., Jensen A., Speyer D., Sturmfels B., Thomas R.: Computing tropical varieties. J. Symbolic Comput. 42(1), 54–73 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bridson M.R., Häfliger A.: Metric Spaces of Non-Positive Curvature. Springer-Verlag, Berlin (1999)zbMATHGoogle Scholar
  4. 4.
    Brown K.S.: Buildings. Springer-Verlag, Berlin/New York (1989)zbMATHGoogle Scholar
  5. 5.
    Bruhat F., Tits J.: Schémas en groupes et immeubles des groupes classiques sur un corps local. Bull. Soc. Math. France 112, 259–301 (1984)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Bruhat F., Tits J.: Errata: Group schemes and buildings of classical groups over a local field. Bull. Soc. Math. France 115(2), 194 (1987)MathSciNetGoogle Scholar
  7. 7.
    Cohen I.S.: On non-Archimedean normed spaces. Indag. Math. 10, 244–249 (1948)Google Scholar
  8. 8.
    Develin M., Sturmfels B.: Tropical convexity. Doc. Math. 9, 1–27 (2004)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Dress A.: Trees, tight extensions of metric spaces and the cohomological dimension of certain groups: a note on combinatorial properties of metric spaces. Adv. Math. 53(3), 321–402 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Dress A., Moulton V., Terhalle W.: T-theory—an overview. European J. Combin. 17(2-3), 161–175 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Dress A., Terhalle W.: A combinatorial approach to \({\wp}\)-adic geometry. Geom. Dedicata 46(2), 127–148 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Dress A., Terhalle W.: The tree of life and other affine buildings. Doc. Math. Extra Vol. III, 565–574 (1998)Google Scholar
  13. 13.
    Dress A.: Duality theory for finite and infinite matroids with coefficients. Adv. Math. 59(2), 97–123 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Dress A., Wenzel W.: Valuated matroids: a new look at the greedy algorithm. Appl. Math. Lett. 3(2), 33–35 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Dress A., Wenzel W.: A greedy-algorithm characterization of valuated Δ-matroids. Appl. Math. Lett. 4(6), 55–58 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Dress A., Wenzel W.: Valuated matroids. Adv. Math. 93(2), 214–250 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Garret P.: Buildings and Classical Groups. Chapman & Hall, London (1997)Google Scholar
  18. 18.
    Goldman O., Iwahori N.: The space of \({\mathfrak p}\)-adic norms. Acta Math. 109, 137–177 (1963)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Gromov, M.: Hyperbolic groups. In: Gerstin, S. (ed.) Essays in Group Theory, pp. 75–263. Springer-Verlag, New York (1987)Google Scholar
  20. 20.
    Joswig M., Sturmfels B., Yu J.: Affine buildings and tropical convexity. Albanian J. Math. 1(4), 187–211 (2007)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Monna, A.F.: Sur les espaces linéaires normés I-IV. Indag. Math. 8, 643–653, 654–660, 632–689, 690–700 (1946)Google Scholar
  22. 22.
    Murota K.: Matrices and Matroids for Systems Analysis. Springer-Verlag, Berlin (2000)zbMATHGoogle Scholar
  23. 23.
    Ronan M.: Lectures on Buildings. Academic Press, Boston (1989)zbMATHGoogle Scholar
  24. 24.
    Speyer D., Sturmfels B.: The tropical Grassmannian. Adv. Geom. 4(3), 389–411 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Terhalle, W.: Ein kombinatorischer Zugang zu \({\wp}\)-adischer Geometrie: Bewertete Matroide, Bäume und Gebäude. Dissertation, Universität Bielefeld, Bielefeld (1992)Google Scholar
  26. 26.
    Terhalle, W.: Matroidal trees: a unifying theory of treelike spaces and their ends. Habilitationsschrift, Universität Bielefeld, Bielefeld (1998)Google Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of Combinatorics and GeometryCAS-MPG Partner Institute for Computational Biology, Shanghai Institutes for Biological Sciences, Chinese Academy of SciencesShanghaiChina
  2. 2.Department of MathematicsUppsala University, BMCUppsalaSweden
  3. 3.School of Computing SciencesUniversity of East AngliaNorwichUK

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