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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[edit]

A finite group $G$ 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 $G$ . 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[edit]

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

For any chain $\mathbf {c}$ of elements, the smallest sublattice of $L$ containing all the elements of $\mathbf {m}$ and $\mathbf {c}$ is distributive.^[4] This is the original condition of Stanley.^[2]
Every element of $\mathbf {m}$ is left modular. That is, for each $m$ in $\mathbf {m}$ and each $x\leq y$ in $L$ , we have $(x\vee m)\wedge y=x\vee (m\wedge y).$ ^[5]^[6]
Every element of $\mathbf {m}$ is rank modular, in the following sense: if $\rho$ is the rank function of $L$ , then for each $m$ in $\mathbf {m}$ and each $x$ in $L$ , we have $\rho (m\wedge x)+\rho (m\vee x)=\rho (m)+\rho (x).$ ^[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[edit]

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[edit]

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 $n$ is supersolvable if and only if it has an edge lexicographic labeling that assigns to each maximal chain a permutation of $\{1,\dots ,n\}.$ ^[18]

Notes[edit]

^ Schmidt (1994, Theorem 2.1.3 and surrounding discussion)
^ ^a ^b Stanley (1972)
^ ^a ^b ^c ^d Stern (1999, p. 162)
^ Stern (1999, Section 4.3)
^ Stern (1999, Corollary 4.3.3) (for semimodular lattices)
^ McNamara & Thomas (2006, Theorem 1)
^ Stanley (2007, Proposition 4.10) (for geometric lattices)
^ Foldes & Woodroofe (2021, Theorem 1.4)
^ Stern (1999, Theorems 1.72 and 1.73)
^ McNamara & Thomas (2006)
^ Heller & Schwer (2018)
^ Simion (2000, p. 370)
^ Stanley (2007, Corollary 4.10)
^ Sagan (1999, Section 6)
^ Stanley (2007, Corollary 4.9)
^ Stanley (2007, Theorem 4.13)
^ Yuzvinsky (2001, Section 6.3)
^ McNamara & Thomas (2006, p. 101)

References[edit]

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

[1] Schmidt (1994, Theorem 2.1.3 and surrounding discussion)

[origpaper-2] Stanley (1972)

[sternexamples-3] 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)

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