Supersolvable Lattices Research Articles

A modular characterization of supersolvable lattices

We characterize supersolvable lattices in terms of a certain modular type relation. McNamara and Thomas earlier characterized this class of lattices as those graded lattices having a maximal chain that consists of left-modular elements. Our characterization replaces the condition of gradedness with a second modularity condition on the maximal chain of left-modular elements.

Two Posets of Noncrossing Partitions Coming From Undesired Parking Spaces

Considere las particiones sin cruces de un conjunto de n elementos que no usan el bloque {n - 1, n}, ni usan a la vez el bloque {n} y un bloque que contenga a 1 y n - 1. En este artículo estudiamos el subposet del retículo de particiones sin cruces inducido por estos elementos. Probamos que este retículo es supersoluble, y por lo tanto es lexicogríaficamente descascarable. También damos un modelo combinatorio de las bases NBB de este retículo y derivamos una fórmula explicita para el valor de su función de Möbius entre el elemento mínimo y el máximo. Este trabajo es motivado por un artículo reciente de M. Bruce, M. Dougherty, M. Hlavacek, R. Kudo, e I. Nicolas en el cual introducen un subposet del retículo de particiones sin cruces que es determinado por funciones de parqueo con ciertas entradas prohibidas. En particular, ellos conjeturan que el poset resultante siempre tiene un complejo de orden contráctil. En este artículo probamos esta conjetura, sumergiendo su poset en el nuestro y mostrando que esta inmersión hereda la descascarabilidad lexicográfica.

Two Posets of Noncrossing Set Partitions

In this manuscript we consider a subposet of the lattice of noncrossing set partitions of an n-element set under refinement order. This subposet is induced by those noncrossing set partitions, which do not contain the block {n−1,n}, or which do not contain the singleton block {n} whenever 1 and n−1 are in the same block. We prove that the resulting poset is in fact a supersolvable lattice, and we give a combinatorial proof for the value of its Möbius function between least and greatest element by using Blass and Sagan's theory of NBB bases.As a corollary we prove a conjecture by Bruce, Dougherty, Hlavacek, Kudo and Nicolas about the homotopy type of a certain subposet of our poset, which comes from parking functions with undesired positions.

A broad class of shellable lattices

We introduce a new class of lattices, the modernistic lattices, and their duals, the comodernistic lattices. We show that every modernistic or comodernistic lattice has shellable order complex. We go on to exhibit a large number of examples of (co)modernistic lattices. We show comodernism for two main families of lattices that were not previously known to be shellable: the order congruence lattices of finite posets, and a weighted generalization of the k-equal partition lattices.We also exhibit many examples of (co)modernistic lattices that were already known to be shellable. To begin with, the definition of modernistic is a common weakening of the definitions of semimodular and supersolvable. We thus obtain a unified proof that lattices in these classes are shellable.Subgroup lattices of solvable groups form another family of comodernistic lattices that were already proven to be shellable. We show not only that subgroup lattices of solvable groups are comodernistic, but that solvability of a group is equivalent to the comodernistic property on its subgroup lattice. Indeed, the definition of comodernistic exactly requires on every interval a lattice-theoretic analogue of the composition series in a solvable group. Thus, the relation between comodernistic lattices and solvable groups resembles, in several respects, that between supersolvable lattices and supersolvable groups.

A lexicographic shellability characterization of geometric lattices

Geometric lattices are characterized in this paper as those finite, atomic lattices such that every atom ordering induces a lexicographic shelling given by an edge labeling known as a minimal labeling. Equivalently, geometric lattices are shown to be exactly those finite lattices such that every ordering on the join-irreducibles induces a lexicographic shelling. This new characterization fits into a similar paradigm as McNamaraʼs characterization of supersolvable lattices as those lattices admitting a different type of lexicographic shelling, namely one in which each maximal chain is labeled with a permutation of {1,…,n}.

Antichain Cutsets of Strongly Connected Posets

Rival and Zaguia showed that the antichain cutsets of a finite Boolean lattice are exactly the level sets. We show that a similar characterization of antichain cutsets holds for any strongly connected poset of locally finite height. As a corollary, we get such a characterization for semimodular lattices, supersolvable lattices, Bruhat orders, locally shellable lattices, and many more. We also consider a generalization to strongly connected hypergraphs having finite edges.

Irreducible representations of semisimple algebraic groups and supersolvable lattices

Let Λ be the cross section lattice of an irreducible representation of a semisimple algebraic group. Certain combinatorial properties of Λ are studied. Supersolvable Λʼs are determined in terms of Dynkin diagrams. The characteristic polynomial of a supersolvable Λ is computed.

A Convex-Ear Decomposition for Rank-Selected Subposets of Supersolvable Lattices

Let $L$ be a supersolvable lattice with nonzero Mobius function. We show that the order complex of any rank-selected subposet of $L$ admits a convex-ear decomposition. This proves many new inequalities for the h-vectors of such complexes, and shows that their g-vectors are M-vectors.

Supersolvable LL-lattices of binary trees

Some posets of binary leaf-labeled trees are shown to be supersolvable lattices and explicit EL-labelings are given. Their characteristic polynomials are computed, recovering their known factorization in a different way.

Connectivity of h-complexes

This paper verifies a conjecture of Edelman and Reiner regarding the homology of the h-complex of a Boolean algebra. A discrete Morse function with no low-dimensional critical cells is constructed, implying a lower bound on connectivity. This together with an Alexander duality result of Edelman and Reiner implies homology vanishing also in high dimensions. Finally, possible generalizations to certain classes of supersolvable lattices are suggested.

Why the characteristic polynomial factors

We survey three methods for proving that the characteristic polynomial of a finite ranked lattice factors over the nonnegative integers and indicate how they have evolved recently. The first technique uses geometric ideas and is based on Zaslavsky’s theory of signed graphs. The second approach is algebraic and employs results of Saito and Terao about free hyperplane arrangements. Finally we consider a purely combinatorial theorem of Stanley about supersolvable lattices and its generalizations.

Möbius Functions of Lattices

We introduce the concept of a bounded below set in a lattice. This can be used to give a generalization of Rota's broken circuit theorem to any finite lattice. We then show how this result can be used to compute and combinatorially explain the Möbius function in various examples including non-crossing set partitions, shuffle posets, and integer partitions in dominance order. Next we present a generalization of Stanley's theorem that the characteristic polynomial of a semimodular supersolvable lattice factors over the integers. We also give some applications of this second main theorem, including the Tamari lattices.

A generalization of semiconductor supersolvable lattices

Stanley ( Algebra Universalis 2, 1972, 197–217) introduced the notion of a supersolvable lattice, L, in part to combinatorially explain the factorization of its characteristic polynomial over the integers when L is also semimodular. He did this by showing that the roots of the polynomial count certain sets of atoms of the lattice. In the present work we define an object called an atom decision tree. The class of semimodular lattices with atom decision trees strictly contains the class of supersolvable lattices, but their characteristic polynomials still factor for combinatorial reasons. We then apply this notion to prove the factorization of polynomials associated with various hyperplane arrangements having non-supersolvable lattices.

On the Cohen-Macaulay connectivity of supersolvable lattices and the homotopy type of posets

It is a well known fact that a supersolvable lattice is EL-shellable. Hence a supersolvable lattice (resp., its Stanley-Reisner ring) is Cohen-Macaulay. We prove that if L is a supersolvable lattice such that all intervals have non-vanishing Möbius number, then for an arbitrary element x ϵ L the poset L − { x} is also Cohen-Macaulay. Posets with this property are called 2-Cohen-Macaulay posets. In particular, in this case the type of the Stanley-Reisner ring of L is given by the absolute value of the Möbius number μ( L). On the other hand, it is a simple observation that the non-vanishing of the Möbius number on intervals is a necessary condition for a poset to be 2-Cohen-Macaulay. For the proof of the 2-Cohen-Macaulayness we will derive some remarkable results about the homotopy type of posets, in particular on posets the homotopy type of which is a wedge of spheres.

Whitney numbers of some geometric lattices

It is a well known fact that the Whitney numbers of a sequence of ranked posets P0, P1, P2,-.of rank 0, l, 2 .... such that every r-ranked filter of P, is isomorphic to Pr are the connection constants between the sequence of powers and the sequence of the characteristic polynomials of the posets [-Dow]. We note here that, if the posets are indeed supersolvable geometric lattices, then the sequence of characteristic polynomials is persistent in the sense of [Dam2]. Thus, in Sections 3 and 4, we will be able to transfer the two term recursion and the explicit formula for connection constants [Daml] as well as several log-concavity properties of symmetric functions [Sag2] to the Whitney numbers of those lattices. Moreover, since the roots of the characteristic polynomial of a supersolvable lattice have been given a nice combinatorial meaning [Sta2], we can also give a simple semantics for those formulas. As a consequence of our results, we obtain (Section 5) unifying proofs of several properties enjoyed by Whitney numbers of Boolean algebras, subspace lattices, partition lattices, and Dowling lattices. It turns out that these lattices form the only infinite sequences of modularly complemented geometric lattices satisfying the conditions mentioned at the beginning.

The lattice of closure relations on a poset

In this paper we show that the set of closure relations on a finite posetP forms a supersolvable lattice, as suggested by Rota. Furthermore this lattice is dually isomorphic to the lattice of closed sets in a convex geometry (in the sense of Edelman and Jamison [EJ]). We also characterize the modular elements of this lattice (whenP has a greatest element) and compute its characteristic polynomial.

Free arrangements of hyperplanes and supersolvable lattices

A finite family of hyperplanes through the origin in C’+’ is called an Iarrangement of hyperplanes, or simply an l-arrangement. An l-arrangement is said to be free if the module of logarithmic vector fields (=holomorphic vector fields tangent to all hyperplanes in the arrangement) is a free module (the accurate definition will be given in (2.1)). Let A be a free I-arrangement. Then a set {do,..., d,} of I + 1 nonnegative integers can be defined (2.3). These (d, ,..., d,) are called the exponents of A. On the other hand, one can associate a geometric lattice L(A) with any arrangement A (3.1). The characteristic polynomial XLca,(t) E Z[t] is defined by using the Mobius function (3.4). The second author [7] proved that

Shellable and Cohen-Macaulay partially ordered sets

In this paper we study shellable posets (partially ordered sets), that is, finite posets such that the simplicial complex of chains is shellable. It is shown that all admissible lattices (including all finite semimodular and supersolvable lattices) and all bounded locally semimodular finite posets are shellable. A technique for labeling the edges of the Hasse diagram of certain lattices, due to R. Stanley, is generalized to posets and shown to imply shellability, while Stanley’s main theorem on the Jordan-HÖlder sequences of such labelings remains valid. Further, we show a number of ways in which shellable posets can be constructed from other shellable posets and complexes. These results give rise to several new examples of Cohen-Macaulay posets. For instance, the lattice of subgroups of a finite group G is Cohen-Macaulay (in fact shellable) if and only if G is supersolvable. Finally, it is shown that all the higher order complexes of a finite planar distributive lattice are shellable.

Finite lattices and Jordan-Hölder sets

In this paper we extend some aspects of the theory of 'supersolvable lattices' [3] to a more general class of finite lattices which includes the upper-semimodular lattices. In particular, all conjectures made in [33 concerning upper-semimodular lattices will be proved. For instance, we will prove that if L is finite upper-semimodular and if L' denotes L with any set of 'levels' removed, then the M6bius function of L' alternates in sign. Familiarity with [3] will be helpful but not essential for the understanding of the results of this paper. However, many of the proofs are identical to the proofs in [3-I (once the machinery has been suitably generalized) and will be omitted.

Supersolvable lattices

Supersolvable lattices