Minimal Paths between Maximal Chains in Finite Rank Semimodular Lattices
David Samuel Herscovici
DOI: 10.1023/A:1008615111070
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.
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.Pages: 17–37
Keywords: semimodular lattice; maximal chain; Jordan-Hölder permutation; reduced decomposition
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References
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2. H. Abels, “The geometry of the chamber system of a semimodular lattice,” Order 8 (1991), 143-158.
3. G. Birkhoff, Lattice Theory, third edition, American Mathematical Society, Providence, RI, 1967.
4. A. Bj\ddot orner, “Shellable and Cohen-Macaulay partially ordered sets,” Transactions of the American Mathematical Society 260 (1980), 159-183.
5. H. Crapo and G.-C. Rota, On the Foundations of Combinatorial Theory: Combinatorial Geometries, M.I.T. Press, Cambridge, MA, 1970.
6. P. Crawley and R.P. Dilworth, Algebraic Theory of Lattices, Prentice-Hall Inc., Englewood Cliffs, NJ, 1973.
7. G. Gr\ddot atzer, General Lattice Theory, Birkh\ddot auser Verlag, Basel, Germany, 1978.
8. D.S. Herscovici, “Semimodular Lattices and Semibuildings,” J. Alg. Combin. 7 (1998), 39-51.
9. M. Ronan, Lectures on Buildings, Harcourt Brace Jovanovich, Boston, MA, 1989.
10. R.P. Stanley, Enumerative Combinatorics, Wadsworth & Brooks/Cole, Belmont, CA, 1986, Vol. I.
11. R.P. Stanley, “Supersolvable lattices,” Algebra Universalis 2 (1972), 197-217.
12. R.P. Stanley, “Finite lattices and Jordan-H\ddot older sets,” Algebra Universalis 4 (1974), 361-371.