Macaulay Posets Research Articles

The colored symmetric and exterior algebras

We study colored generalizations of the symmetric algebra and its Koszul dual, the exterior algebra. The symmetric group Sn acts on the multilinear components of these algebras. While Sn acts trivially on the multilinear components of the colored symmetric algebra, we use poset topology techniques to understand the representation on its Koszul dual. We introduce an Sn-poset of weighted subsets that we call the weighted boolean algebra and we prove that the multilinear components of the colored exterior algebra are Sn-isomorphic to the top cohomology modules of its maximal intervals. We use a technique of Sundaram to compute group representations on Cohen–Macaulay posets to give a generating formula for the Frobenius series of the colored exterior algebra. We exploit that formula to find an explicit expression for the expansion of the corresponding representations in terms of irreducible Sn-representations. We show that the two colored Koszul dual algebras are Koszul in the sense of Priddy.

Weakly Cohen–Macaulay posets and a class of finite-dimensional graded quadratic algebras

To a finite ranked poset Γ we associate a finite-dimensional graded quadratic algebra RΓ. Assuming Γ satisfies a combinatorial condition known as uniform, RΓ is related to a well-known algebra, the splitting algebraAΓ. First introduced by Gelfand, Retakh, Serconek, and Wilson, splitting algebras originated from the problem of factoring non-commuting polynomials. Given a finite ranked poset Γ, we ask: Is RΓ Koszul? The Koszulity of RΓ is related to a combinatorial topology property of Γ called Cohen–Macaulay. Kloefkorn and Shelton proved that if Γ is a finite ranked cyclic poset, then Γ is Cohen–Macaulay if and only if Γ is uniform and RΓ is Koszul. We define a new generalization of Cohen–Macaulay, weakly Cohen–Macaulay. This new class includes non-uniform posets and posets with disconnected open subintervals. Using a spectral sequence associated to Γ and the notion of a noncommutative Koszul filtration for RΓ, we prove: if Γ is a finite ranked cyclic poset, then Γ is weakly Cohen–Macaulay if and only if RΓ is Koszul. In addition, we prove that Γ is Cohen–Macaulay if and only if Γ is uniform and weakly Cohen–Macaulay.

On the Möbius function of a lower Eulerian Cohen–Macaulay poset

A certain inequality is shown to hold for the values of the Mobius function of the poset obtained by attaching a maximum element to a lower Eulerian Cohen---Macaulay poset. In two important special cases, this inequality provides partial results supporting Stanley's nonnegativity conjecture for the toric h-vector of a lower Eulerian Cohen---Macaulay meet-semilattice and Adin's nonnegativity conjecture for the cubical h-vector of a Cohen---Macaulay cubical complex.

Torus graphs and simplicial posets

For several important classes of manifolds acted on by the torus, the information about the action can be encoded combinatorially by a regular n-valent graph with vector labels on its edges, which we refer to as the torus graph. By analogy with the GKM-graphs, we introduce the notion of equivariant cohomology of a torus graph, and show that it is isomorphic to the face ring of the associated simplicial poset. This extends a series of previous results on the equivariant cohomology of torus manifolds. As a primary combinatorial application, we show that a simplicial poset is Cohen–Macaulay if its face ring is Cohen–Macaulay. This completes the algebraic characterisation of Cohen–Macaulay posets initiated by Stanley. We also study blow-ups of torus graphs and manifolds from both the algebraic and the topological points of view.

Macaulay Posets

Macaulay posets are posets for which there is an analogue of the classical Kruskal-Katona theorem for finite sets. These posets are of great importance in many branches of combinatorics and have numerous applications. We survey mostly new and also some old results on Macaulay posets, where the intention is to present them as pieces of a general theory. In particular, the classical examples of Macaulay posets are included as well as new ones. Emphasis is also put on the construction of Macaulay posets, and their relations to other discrete optimization problems.

Bier Spheres and Posets

In 1992 Thomas Bier presented a strikingly simple method to produce a huge number of simplicial (n – 2)-spheres on 2n vertices, as deleted joins of a simplicial complex on n vertices with its combinatorial Alexander dual.Here we interpret his construction as giving the poset of all the intervals in a boolean algebra that “cut across an ideal.” Thus we arrive at a substantial generalization of Bier’s construction: the Bier posets Bier(P, I) of an arbitrary bounded poset P of finite length. In the case of face posets of PL spheres this yields cellular “generalized Bier spheres.” In the case of Eulerian or Cohen–Macaulay posets P we show that the Bier posets Bier(P, I) inherit these properties.In the boolean case originally considered by Bier, we show that all the spheres produced by his construction are shellable, which yields “many shellable spheres,” most of which lack convex realization. Finally, we present simple explicit formulas for the g-vectors of these simplicial spheres and verify that they satisfy a strong form of the g-conjecture for spheres.

Nonexistence of a Kruskal-Katona Type Theorem for Subword Orders

We consider the poset SO(n) of all words over an n-element alphabet ordered by the subword relation. It is known that SO(2) falls into the class of Macaulay posets, i. e. there is a theorem of Kruskal–Katona type for SO(2). As the corresponding linear ordering of the elements of SO(2) the vip-order can be chosen.Daykin introduced the V-order which generalizes the vip-order to the n≥2 case. He conjectured that the V-order gives a Kruskal–Katona type theorem for SO(n).We show that this conjecture fails for all n≥3 by explicitly giving a counterexample. Based on this, we prove that for no n≥3 the subword order SO(n) is a Macaulay poset.

A new approach to Macaulay posets

We develop a new approach for establishing the Macaulayness of posets representable as cartesian powers of other posets. This approach is based on a problem of constructing an ideal of maximum rank in a poset. Using the relations between the maximum rank ideal problem and the edge-isoperimetric problem on graphs we demonstrate an application of our approach to specification of all posets with a special Macaulay order. We also present a new general construction for additive Macaulay posets and introduce several new families of Macaulay posets.

On the orthogonal product of simplices and direct products of truncated Boolean lattices

The initial point of this paper are two Kruskal–Katona type theorems. The first is the Colored Kruskal–Katona Theorem which can be stated as follows: Direct products of the form B k 1 1× B k 2 1×⋯× B k n 1 belong to the class of Macaulay posets, where B k t denotes the poset consisting of the t+1 lowest levels of the Boolean lattice B k . The second one is a recent result saying that also the products B k 1 k 1−1 × B k 2 k 2−1 ×⋯× B k n k n −1 are Macaulay posets. The main result of this paper is that the natural common generalization to products of truncated Boolean lattices does not hold, i.e. that ( B k t ) n is a Macaulay poset only if t∈{0,1, k−1, k}.

A local–global principle for vertex-isoperimetric problems

We consider the vertex-isoperimetric problem (VIP) for cartesian powers of a graph G. A total order ≼ on the vertex set of G is called isoperimetric if the boundary of sets of a given size k is minimum for any initial segment of ≼, and the ball around any initial segment is again an initial segment of ≼. We prove a local–global principle with respect to the so-called simplicial order on G n (see Section 2 for the definition). Namely, we show that the simplicial order ≼ n is isoperimetric for each n⩾1 iff it is so for n=1,2. Some structural properties of graphs that admit simplicial isoperimetric orderings are presented. We also discuss new relations between the VIP and Macaulay posets.

Another Generalization of Lindström's Theorem on Subcubes of a Cube

We consider the poset P( N; A 1, A 2,…, A m ) consisting of all subsets of a finite set N which do not contain any of the A i 's, where the A i 's are mutually disjoint subsets of N. The elements of P are ordered by inclusion. We show that P belongs to the class of Macaulay posets, i.e. we show a Kruskal–Katona-type theorem for P. For the case that the A i 's form a partition of N, the dual P * of P came to be known as the orthogonal product of simplices. Since the property of being a Macaulay poset is preserved by turning to the dual, we show, in particular, that orthogonal products of simplices are Macaulay posets. Besides, we prove that the posets P and P * are additive.

Log-concavity and compressed ideals in certain Macaulay posets

Let B n be the poset of subsets of {1,2,…, n} ordered by inclusion and M n be the poset of monomials in x 1, x 2,…, x n ordered by divisibility. Both these posets have an additional linear order making them what is called Macaulay posets. We show in this paper that the profiles (also called f-vectors) of ideals in B n and M n generated by the first elements (relatively to the linear order) of a given rank are log-concave.

The Spider Poset Is Macaulay

Let Q(k, l) be a poset whose Hasse diagram is a regular spider with k+1 legs having the same length l. We show that for any n⩾1 the nth cartesian power of the spider poset Q(k, l) is a Macaulay poset for any k⩾0 and l⩾1. In combination with our recent results (S. L. Bezrukov, 1998, J. Combin. Theory Ser. A84, 157–170) this provides a complete characterization of all Macaulay posets which are cartesian powers of upper semilattices, whose Hasse diagrams are trees.

A property of colored complexes and their duals

A ranked poset P is a Macaulay poset if there is a linear order ≺ of the elements of P such that for any m, i the set C( m, i) of the m (with respect to ≺) smallest elements of rank i has minimum-sized shadow among all m-element subsets of the ith level, and the shadow of C( m, i) consists of the smallest elements of the ( i−1)th level. P is called shadow-increasing if for all m, i the shadow of C( m, i) is not smaller than the shadow of C( m, i−1). We show that colored complexes and their duals, the star posets, are shadow-increasing.

Characterizing Profiles ofk-Families in Additive Macaulay Posets

A subset of a poset is ak-familyif there is no chain consisting ofk+1 of its elements. A subset of a ranked poset consisting ofpi elements of ranki,i=0,1,...,Ris said to haveprofilep0,p1,…,pR. A characterization is given for profiles ofk-families in additive Macaulay posets.

Otter's Method and the Homology of Homeomorphically Irreduciblek-Trees

Let T(k)ndenote the collection of trees withnk+2 labelled leaves which have the property that every internal node has degreemk+2 for somem⩾1. Partially order the elements of T(k)nby definingT⩽SifTcan be obtained fromSby contracting some collection of internal edges. Then T(k)nis a Cohen–Macaulay poset with an action ofSnk+2. We show that the top homology of T(k)nas anSnk+2-module is[formula]whereLie(k)Nis the action ofSNon the 1N-homogeneous piece of the free Liek-algebra. This generalizes the result obtained by Sarah Whitehouse [W] in the casek=1.