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The geometry of the chamber system of a semimodular lattice

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Abstract

In this paper geometric properties of the following metric space C are studied. Its elements are called chambers and are the maximal chains of a semimodular lattice X of finite height and its metric d is the gallery distance. We show that X has many properties in common with buildings. More specifically, Tits [17] has recently described buildings in terms of “Weyl-group valued distance functions”. We consider the Jordan-Hölder permutation π(C, D) corresponding to a pair C, D of chambers and show that it has most properties of such a distance with values in the symmetric group.

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  1. Fakultät für Mathematik, Universität Bielefeld, Postfach 8640, 4800, Bielefeld 1, Germany

    Herbert Abels

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  1. Herbert Abels
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Communicated by M. Pouzet

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Abels, H. The geometry of the chamber system of a semimodular lattice. Order 8, 143–158 (1991). https://doi.org/10.1007/BF00383400

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  • DOI: https://doi.org/10.1007/BF00383400

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