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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
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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 
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© 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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