Supersolvable lattice

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.{{harvtxt|Schmidt|1994|loc=Theorem 2.1.3 and surrounding discussion}} Richard Stanley noticed in the 1970s that certain geometric lattices, such as the partition lattice, obeyed similar properties, and gave a lattice-theoretic abstraction.{{harvtxt|Stanley|1972}}

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

1. For any chain $\backslash mathbf\{c\}$ of elements, the smallest sublattice of $L$ containing all the elements of $\backslash mathbf\{m\}$ and $\backslash mathbf\{c\}$ is distributive.{{harvtxt|Stern|1999|loc=Section 4.3}} This is the original condition of Stanley.
2. Every element of $\backslash mathbf\{m\}$ is left modular. That is, for each $m$ in $\backslash mathbf\{m\}$ and each $x\; \backslash leq\; y$ in $L$, we have $(x\backslash vee\; m)\backslash wedge\; y=x\backslash vee(m\backslash wedge\; y).${{harvtxt|Stern|1999|loc=Corollary 4.3.3}} (for semimodular lattices){{harvtxt|McNamara|Thomas|2006|loc=Theorem 1}}
3. Every element of $\backslash mathbf\{m\}$ is rank modular, in the following sense: if $\backslash rho$ is the rank function of $L$, then for each $m$ in $\backslash mathbf\{m\}$ and each $x$ in $L$, we have $\backslash rho(m\backslash wedge\; x)+\backslash rho(m\backslash vee\; x)=\backslash rho(m)+\backslash rho(x).${{harvtxt|Stanley|2007|loc=Proposition 4.10}} (for geometric lattices){{harvtxt|Foldes|Woodroofe|2021|loc=Theorem 1.4}}

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.{{harvtxt|Stern|1999|loc=Theorems 1.72 and 1.73}}

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.{{harvtxt|McNamara|Thomas|2006}}

File:Noncrossing partitions 4; Hasse.svg 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.{{harvtxt|Stern|1999|p=162}}

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. The noncrossing partition lattice is similarly supersolvable,{{harvtxt|Heller|Schwer|2018}} although it is not geometric.{{harvtxt|Simion|2000|p=370}}

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.{{harvtxt|Stanley|2007|loc=Corollary 4.10}}

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

A finite matroid with a supersolvable lattice of flats (equivalently, a lattice that is both geometric and supersolvable) has a real-rooted characteristic polynomial.{{harvtxt|Sagan|1999|loc=Section 6}}{{harvtxt|Stanley|2007|loc=Corollary 4.9}} This is a consequence of a more general factorization theorem for characteristic polynomials over modular elements.{{harvtxt|Stanley|2007|loc=Theorem 4.13}}

The Orlik-Solomon algebra of an arrangement of hyperplanes with a supersolvable intersection lattice is a Koszul algebra.{{harvtxt|Yuzvinsky|2001|loc=Section 6.3}} 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 $\backslash \{\; 1,\; \backslash dots,\; n\backslash \}.${{harvtxt|McNamara|Thomas|2006|p=101}}

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Category:Lattice theory

Category:Solvable groups