In mathematics, a supersolvable lattice is a graded lattice that has a maximal chain of elements, each of which obeys a certain modularity relationship. The definition encapsulates many of the nice properties of lattices of subgroups of supersolvable groups.

Motivation

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A finite group   is said to be supersolvable if it admits a maximal chain (or series) of subgroups so that each subgroup in the chain is normal in  . A normal subgroup has been known since the 1940s to be left and (dual) right modular as an element of the lattice of subgroups.[1] Richard Stanley noticed in the 1970s that certain geometric lattices, such as the partition lattice, obeyed similar properties, and gave a lattice-theoretic abstraction.[2][3]

Definition

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A finite graded lattice   is supersolvable if it admits a maximal chain   of elements (called an M-chain or chief chain) obeying any of the following equivalent properties.

  1. For any chain   of elements, the smallest sublattice of   containing all the elements of   and   is distributive.[4] This is the original condition of Stanley.[2]
  2. Every element of   is left modular. That is, for each   in   and each   in  , we have  [5][6]
  3. Every element of   is rank modular, in the following sense: if   is the rank function of  , then for each   in   and each   in  , we have  [7][8]

For comparison, a finite lattice is geometric if and only if it is atomistic and the elements of the antichain of atoms are all left modular.[9]

An extension of the definition is that of a left modular lattice: a not-necessarily graded lattice with a maximal chain consisting of left modular elements. Thus, a left modular lattice requires the condition of (2), but relaxes the requirement of gradedness.[10]

Examples

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Hasse diagram of the noncrossing partition lattice on a 4 element set. The leftmost maximal chain is a chief chain.

A group is supersolvable if and only if its lattice of subgroups is supersolvable. A chief series of subgroups forms a chief chain in the lattice of subgroups.[3]

The partition lattice of a finite set is supersolvable. A partition is left modular in this lattice if and only if it has at most one non-singleton part.[3] The noncrossing partition lattice is similarly supersolvable,[11] although it is not geometric.[12]

The lattice of flats of the graphic matroid for a graph is supersolvable if and only if the graph is chordal. Working from the top, the chief chain is obtained by removing vertices in a perfect elimination ordering one by one.[13]

Every modular lattice is supersolvable, as every element in such a lattice is left modular and rank modular.[3]

Properties

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A finite matroid with a supersolvable lattice of flats (equivalently, a lattice that is both geometric and supersolvable) has a real-rooted characteristic polynomial.[14][15] This is a consequence of a more general factorization theorem for characteristic polynomials over modular elements.[16]

The Orlik-Solomon algebra of an arrangement of hyperplanes with a supersolvable intersection lattice is a Koszul algebra.[17] For more information, see Supersolvable arrangement.

Any finite supersolvable lattice has an edge lexicographic labeling (or EL-labeling), hence its order complex is shellable and Cohen-Macaulay. Indeed, supersolvable lattices can be characterized in terms of edge lexicographic labelings: a finite lattice of height   is supersolvable if and only if it has an edge lexicographic labeling that assigns to each maximal chain a permutation of  [18]

Notes

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  1. ^ Schmidt (1994, Theorem 2.1.3 and surrounding discussion)
  2. ^ a b Stanley (1972)
  3. ^ a b c d Stern (1999, p. 162)
  4. ^ Stern (1999, Section 4.3)
  5. ^ Stern (1999, Corollary 4.3.3) (for semimodular lattices)
  6. ^ McNamara & Thomas (2006, Theorem 1)
  7. ^ Stanley (2007, Proposition 4.10) (for geometric lattices)
  8. ^ Foldes & Woodroofe (2021, Theorem 1.4)
  9. ^ Stern (1999, Theorems 1.72 and 1.73)
  10. ^ McNamara & Thomas (2006)
  11. ^ Heller & Schwer (2018)
  12. ^ Simion (2000, p. 370)
  13. ^ Stanley (2007, Corollary 4.10)
  14. ^ Sagan (1999, Section 6)
  15. ^ Stanley (2007, Corollary 4.9)
  16. ^ Stanley (2007, Theorem 4.13)
  17. ^ Yuzvinsky (2001, Section 6.3)
  18. ^ McNamara & Thomas (2006, p. 101)

References

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  • Foldes, Stephan; Woodroofe, Russ (2021), "A Modular Characterization of Supersolvable Lattices", Proceedings of the American Mathematical Society, 150 (1): 31–39, arXiv:2011.11657, doi:10.1090/proc/15645
  • Heller, Julia; Schwer, Petra (2018), "Generalized Non-crossing Partitions and Buildings", Electronic Journal of Combinatorics, 25 (1), arXiv:1706.00529, doi:10.37236/7200
  • McNamara, Peter; Thomas, Hugh (2006), "Poset Edge-Labellings and Left Modularity", European Journal of Combinatorics, 27 (1): 101–113, arXiv:math.CO/0211126, doi:10.1016/j.ejc.2004.07.010
  • Sagan, Bruce (1999), "Why the characteristic polynomial factors", Bulletin of the American Mathematical Society, 36 (2): 113–133, arXiv:math/9812136, doi:10.1090/S0273-0979-99-00775-2
  • Schmidt, Roland (1994), Subgroup lattices of groups, de Gruyter Expositions in Mathematics, vol. 14, Walter de Gruyter & Co., doi:10.1515/9783110868647, ISBN 3-11-011213-2
  • Simion, Rodica (2000), "Noncrossing Partitions", Discrete Mathematics, Formal Power Series and Algebraic Combinatorics (Vienna 1997), 217 (1–3): 367–409, doi:10.1016/S0012-365X(99)00273-3
  • Stanley, Richard P. (1972), "Supersolvable Lattices", Algebra Universalis, 2: 197–217, doi:10.1007/BF02945028
  • Stanley, Richard P. (2007), "An Introduction to Hyperplane Arrangements", Geometric combinatorics, IAS/Park City Mathematics Series, vol. 13, American Mathematical Society, pp. 389–496, ISBN 978-0-8218-3736-8
  • Stern, Manfred (1999), Semimodular Lattices. Theory and Applications, Encyclopedia of Mathematics and its Applications, vol. 73, Cambridge University Press, doi:10.1017/CBO9780511665578, ISBN 0-521-46105-7
  • Yuzvinsky, Sergey (2001), "Orlik–Solomon algebras in algebra and topology", Russian Mathematical Surveys, 56 (2): 293–364, doi:10.1070/RM2001v056n02ABEH000383, MR 1859708