Supersolvable lattice

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

A finite group ${\displaystyle G}$ 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 ${\displaystyle G}$. 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

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

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

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] thus, the partition lattice is also geometric. ^[11] The noncrossing partition lattice is similarly supersolvable, ^[12] although it is not geometric. ^[13]

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. ^[14]

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

Properties

A finite matroid with a supersolvable lattice of flats (equivalently, a lattice that is both geometric and supersolvable) has a real-rooted characteristic polynomial. ^{[15] [16]} This is a consequence of a more general factorization theorem for characteristic polynomials over modular elements. ^[17]

The Orlik-Solomon algebra of an arrangement of hyperplanes with a supersolvable intersection lattice is a Koszul algebra. ^[18] 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 ${\displaystyle n}$ is supersolvable if and only if it has an edge lexicographic labeling that assigns to each maximal chain a permutation of ${\displaystyle \{1,\dots ,n\}.}$ ^[19]

Notes

1. Schmidt (1994 , Theorem 2.1.3 and surrounding discussion)
2. Stanley (1972)
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. Stanley (2007 , Example 4.11)
12. Heller & Schwer (2018)
13. Simion (2000 , p. 370)
14. Stanley (2007 , Corollary 4.10)
15. Sagan (1999 , Section 6)
16. Stanley (2007 , Corollary 4.9)
17. Stanley (2007 , Theorem 4.13)
18. Yuzvinsky (2001 , Section 6.3)
19. McNamara & Thomas (2006 , p. 101)

References

• 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