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] 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

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

Notes

References