Abstract
An explicit formula for the gallery distance of two maximal flags in a vector space is given. The main tool of the proof is the Jordan-Hölder permutation. The result and its proof hold more generally for any semimodular lattice of finite height and with minor changes also for the distance of two chambers in the Bruhat-Tits building of the general linear group.
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H. Abels (1989) Finiteness properties of certain arithmetic groups in the function field case, to be published in the Israel Journal of Mathematics.
H. Abels (1991) The geometry of the chamber system of a semimodular lattice, Order 8(2), forthcoming.
G. Birkhoff (1967) Lattice Theory, 3rd ed. AMS Coll. Publ. 25, Providence, R.I.
A.Björner (1980) Shellable and Cohen-Macaulay partially ordered sets, Trans. AMS. 260, 159–183.
N. Bourbaki (1968) Groupes et algèbres de Lie, chap. IV–VI, Paris.
K. S. Brown (1988) Buildings, Springer Verlag.
A. W. M.Dress and R.Scharlau (1987) Gated sets in metric spaces, Aequ. math. 34, 112–120.
G.Grätzer (1978) General Lattice Theory, Birkhäuser Basel.
D. Grayson (1982) Finite generation of K-groups of a curve over a finite field (after D. Quillen_, Algebraic K-theory. Proc. Oberwolfach conference 1980. Ed. by R. Keith Dennis, Part I, Springer LN 966, 69–90.
R. P.Stanley (1972) Supersolvable lattices, Algebra universalis 2, 197–217.
J.Tits (1986) Buildings and group amalgamations, Proc. Groups St. Andrews 1985. London MS Lecture Note 121, 111–127.
J. Tits (1988/89) Théorie des groupes. Immeubles jumelés, Résumé de cours au Collège de France.
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Communicated by M. Pouzet
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Abels, H. The gallery distance of flags. Order 8, 77–92 (1991). https://doi.org/10.1007/BF00385816
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DOI: https://doi.org/10.1007/BF00385816