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  • Jordan-Hölder permutation  (2)
  • 1
    Electronic Resource
    Electronic Resource
    Semimodular Lattices and Semibuildings (1998)
    Springer
    Journal of algebraic combinatorics 7 (1998), S. 39-51 
    Details
    ISSN: 1572-9192
    Keywords: semimodular lattice ; chamber system ; Jordan-Hölder permutation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In a ranked lattice, we consider two maximal chains, or “flags” to be i-adjacent if they are equal except possibly on rank i. Thus, a finite rank lattice is a chamber system. If the lattice is semimodular, as noted in [9], there is a “Jordan-Hölder permutation” between any two flags. This permutation has the properties of an Sn-distance function on the chamber system of flags. Using these notions, we define a W-semibuilding as a chamber system with certain additional properties similar to properties Tits used to characterize buildings. We show that finite rank semimodular lattices form an Sn-semibuilding, and develop a flag-based axiomatization of semimodular lattices. We refine these properties to axiomatize geometric, modular and distributive lattices as well, and to reprove Tits' result that Sn-buildings correspond to relatively complemented modular lattices (see [16], Section 6.1.5).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008667127908
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  • 2
    Electronic Resource
    Electronic Resource
    Minimal Paths between Maximal Chains in Finite Rank Semimodular Lattices (1998)
    Springer
    Journal of algebraic combinatorics 7 (1998), S. 17-37 
    Details
    ISSN: 1572-9192
    Keywords: semimodular lattice ; maximal chain ; Jordan-Hölder permutation ; reduced decomposition
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We study paths between maximal chains, or “flags,” in finite rank semimodular lattices. Two flags are adjacent if they differ on at most one rank. A path is a sequence of flags in which consecutive flags are adjacent. We study the union of all flags on at least one minimum length path connecting two flags in the lattice. This is a subposet of the original lattice. If the lattice is modular, the subposet is equal to the sublattice generated by the flags. It is a distributive lattice which is determined by the “Jordan-Hölder permutation” between the flags. The minimal paths correspond to all reduced decompositions of this permutation. In a semimodular lattice, the subposet is not uniquely determined by the Jordan-Hölder permutation for the flags. However, it is a join sublattice of the distributive lattice corresponding to this permutation. It is semimodular, unlike the lattice generated by the two flags, which may not be ranked. The minimal paths correspond to some reduced decompositions of the permutation, though not necessarily all. We classify the possible lattices which can arise in this way, and characterize all possibilities for the set of shortest paths between two flags in a semimodular lattice.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008615111070
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