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Symmetric Chain Decompositions of Linear Lattices

Published online by Cambridge University Press:  01 June 1997

FRANK VOGT
Affiliation:
Fachbereich Mathematik, Technische Hochschule Darmstadt, Schloßgartenstraße 7, D-64289 Darmstadt, Germany (e-mail: vogt@mathematik.th_darmstadt.de)
BERND VOIGT
Affiliation:
Fachbereich Mathematik, Universität Bielefeld, D-33501 Bielefeld, Germany

Abstract

It has been known for several years that the lattice of subspaces of a finite vector space has a decomposition into symmetric chains, i.e. a decomposition into disjoint chains that are symmetric with respect to the rank function of the lattice. This paper gives a positive answer to the long-standing open problem of providing an explicit construction of such a symmetric chain decomposition for a given lattice of subspaces of a finite (dimensional) vector space. The construction is done inductively using Schubert normal forms and results in a bracketing algorithm similar to the well-known algorithm for Boolean lattices.

Type
Research Article
Copyright
1997 Cambridge University Press

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