On sequentially Cohen-Macaulay complexes and posets

The classes of sequentially Cohen-Macaulay and sequentially homotopy Cohen-Macaulay complexes and posets are studied. First, some different versions of the definitions are discussed and the homotopy type is determined. Second, it is shown how various constructions, such as join, product and rank-selection preserve these properties. Third, a characterization of sequential Cohen-Macaulayness for posets is given. Finally, in an appendix we outline connections with ring-theory and survey some uses of sequential Cohen-Macaulayness in commutative algebra. Supported in part by National Science Foundation grants DMS 0302310 and DMS 0604562. Supported by Deutsche Forschungsgemeinschaft (DFG)     

Koszul incidence algebras, affine semigroups, and Stanley-Reisner ideals

We prove a theorem unifying three results from combinatorial homological and commutative algebra, characterizing the Koszul property for incidence algebras of posets and affine semigroup rings, and characterizing linear resolutions of squarefree monomial ideals. The characterization in the graded setting is via the Cohen-Macaulay property of certain posets or simplicial complexes, and in the more general nongraded setting, via the sequential Cohen-Macaulay property.     

A Poset Fiber Theorem for Doubly Cohen-Macaulay Posets and Its Applications

This paper studies topological properties of the lattices of non-crossing partitions of types A and B and of the poset of injective words. Specifically, it is shown that after the removal of the bottom and top elements (if existent) these posets are doubly Cohen-Macaulay. This strengthens the well-known facts that these posets are Cohen-Macaulay. Our results rely on a new poset fiber theorem which turns out to be a useful tool to prove double (homotopy) Cohen- Macaulayness of a poset. Applications to complexes of injective words are also included.     

Whitney Homology of Semipure Shellable Posets

We generalize results of Calderbank, Hanlon and Robinson on the representation of the symmetric group on the homology of posets of partitions with restricted block size. Calderbank, Hanlon and Robinson consider the cases of block sizes that are congruent to 0 mod d and 1 mod d for fixed d. We derive a general formula for the representation of the symmetric group on the homology of posets of partitions whose block sizes are congruent to k mod d for any k and d. This formula reduces to the Calderbank-Hanlon-Robinson formulas when k = 0, 1 and to formulas of Sundaram for the virtual representation on the alternating sum of homology. Our results apply to restricted block size partition posets even more general than the k mod d partition posets. These posets include the lattice of partitions whose block sizes are bounded from below by some fixed k. Our main tools involve the new theory of nonpure shellability developed by Björner and Wachs and a generalization of a technique of Sundaram which uses Whitney homology to compute homology representations of Cohen-Macaulay posets. An application to subspace arrangements is also discussed.     

Homology of generalized partition posets

We define a family of posets of partitions associated to an operad. We prove that the operad is Koszul if and only if the posets are Cohen-Macaulay. On the one hand, this characterization allows us to compute completely the homology of the posets. The homology groups are isomorphic to the Koszul dual cooperad. On the other hand, we get new methods for proving that an operad is Koszul.     

Combinatorial methods in the theory of Cohen-Macaulay rings

Characterizations of Cohen-Macaulay posets are given in terms of the nonsingularity of certain incidence matrices. These results are applied to derive a purely combinatorial construction of basic systems for Stanley-Reisner rings of shellable posets. A procedure for transferring identities from one polynomial ring to another is then used to obtain basic systems for partition rings.     

The lexicographic sum of Cohen-Macaulay and shellable ordered sets

We study the problem of determining when the lexicographic sum ∑ _q∈Q P _q of a family of posets {P _q/qεQ} over a posetQ is Cohen-Macaulay or shellable. Our main result, a characterization of when the lexicographic sum is Cohen-Macaulay, is proven using combinatorial methods introduced by Garsia. A similar characterization for when the lexicographic sum is CL (chainwise-lexicographically)-shellable, is derived using the recursive atom ordering method due to Björner and Wachs.     

On Cohen-Macaulay Posets, Koszul Algebras and Certain Modules Associated to Schubert Varieties

In the first part of this paper, some cohomological propertiesof the incidence algebra A_k(X) of a finite poset X over a fieldk are studied. In particular, it is shown that A_k(X) is Koszulif and only if every open interval of X is Cohen-Macaulay overk. In the second part, certain categories of representationsof a Borel subgroup B of a semisimple connected algebraic groupG are studied. It is shown that these categories have the samecohomological properties as certain Cohen-Macaulay posets relatedto the Weyl group of G, and the higher Ext-groups between certainB-modules are then computed, using results from the first part.     

Homotopy type of posets and lattice complementation

This paper is concerned with homotopy properties of partially ordered sets, in particular contractibility. The main result is that a noncomplemented lattice with deleted bounds is contractible. The paper also presents (i) the homology of final sets and cutsets, (ii) a generalization to posets of Rota's crosscut theorem, (iii) contractibility proofs for some classes of posets of interest in fixed point theory, and (iv) a simple characterization of the Cohen-Macaulay property for dismantlable lattices.     

Z-Semicontinuous Posets

Z-semicontinuous posets include semicontinuous lattices and Z-continuous posets as special cases. We characterized when the associated Z-waybelow relation is multiplicative and also make a topological connection.     

3-Interval irreducible partially ordered sets

In this paper we discuss the characterization problem for posets of interval dimension at most 2. We compile the minimal list of forbidden posets for interval dimension 2. Members of this list are called 3-interval irreducible posets. The problem is related to a series of characterization problems which have been solved earlier. These are: The characterization of planar lattices, due to Kelly and Rival [5], the characterization of posets of dimension at most 2 (3-irreducible posets) which has been obtained independently by Trotter and Moore [8] and by Kelly [4] and the characterization of bipartite 3-interval irreducible posets due to Trotter [9].We show that every 3-interval irreducible poset is a reduced partial stack of some bipartite 3-interval irreducible poset. Moreover, we succeed in classifying the 3-interval irreducible partial stacks of most of the bipartite 3-interval irreducible posets. Our arguments depend on a transformationP B(P), such that IdimP=dimB(P). This transformation has been introduced in [2].Supported by the DFG under grant FE 340/2–1.     

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.     

The zeta function of a simplicial complex

Given a simplicial complex δ on vertices {1, …,n} and a fieldF we consider the subvariety of projective (n−1)-space overF consisting of points whose homogeneous coordinates have support in δ. We give a simple rational expression for the zeta function of this singular projective variety overF _q and show a close connection with the Betti numbers of the corresponding variety over ℂ. This connection is particularly simple in the case when Δ is Cohen-Macaulay.     

The rank-generating functions of upho posets

Upper homogeneous finite type (upho) posets are a large class of partially ordered sets with the property that the principal order filter at every vertex is isomorphic to the whole poset. Well-known examples include k-ary trees, the grid graphs, and the Stern poset. Very little is known about upho posets in general. In this paper, we construct upho posets with Schur-positive Ehrenborg quasisymmetric functions, whose rank-generating functions have rational poles and zeros. We also categorize the rank-generating functions of all planar upho posets. Finally, we prove the existence of an upho poset with an uncomputable rank-generating function.     

A Rational-Function Identity Related to the Murnaghan–Nakayama Formula for the Characters of S n

The Murnaghan–Nakayama formula for the characters of S _n is derived from Young's seminormal representation, by a direct combinatorial argument. The main idea is a rational function identity which when stated in a more general form involves Möbius functions of posets whose Hasse diagrams have a planar embedding. These ideas are also used to give an elementary exposition of the main properties of Young's seminormal representations.     

Modular posets and semigroups

Posets and poset homomorphisms (preserving both order and parallelism) have been shown to form a category which is equivalent to the category of pogroupoids and their homomorphisms. Among the posets those posets whose associated pogroupoids are semigroups are identified as being precisely those posets which are (C ₂+1)-free. In the case of lattices this condition means that the lattice is alsoN ₅-free and hence modular. Using the standard connection: semigroup to poset to pogroupoid, it is observed that in many cases the image pogroupoid obtained is a semigroup even if quite different from the original one. The nature of this mapping appears intriguing in the poset setting and may well be so seen from the semigroup theory viewpoint.     

Stable representations of posets

The purpose of this paper is to study stable representations of partially ordered sets (posets) and compare it to the well known theory for quivers. In particular, we prove that every indecomposable representation of a poset of finite type is stable with respect to some weight and construct that weight explicitly in terms of the dimension vector. We show that if a poset is primitive then Coxeter transformations preserve stable representations. When the base field is the field of complex numbers we establish the connection between the polystable representations and the unitary χ-representations of posets. This connection explains the similarity of the results obtained in the series of papers.     

Further results on planar DO functions and commutative semifields

It is proven that any Dembowski–Ostrom polynomial is planar if and only if its evaluation map is 2-to-1, which can be used to explain some known planar Dembowski–Ostrom polynomials. A direct connection between a planar Dembowski–Ostrom polynomial and a permutation polynomial is established if the corresponding semifield is of odd dimension over its nucleus. In addition, all commutative semifields of order 3⁵ are classified.     

Polygon Posets and the Weak Order of Coxeter Groups

We explore the connection between polygon posets, which is a class of ranked posets with an edge-labeling which satisfies certain polygon properties, and the weak order of Coxeter groups. We show that every polygon poset is isomorphic to a join ideal in the weak order, and for Coxeter groups where no pair of generators have infinite order the converse is also true.The class of polygon posets is seen to include the class of generalized quotients defined by Björner and Wachs, while itself being included in the class of alternative generalized quotients also considered by these authors. By studying polygon posets we are then able to answer an open question about common properties of these two classes.     

N-free posets as generalizations of series-parallel posets

N-Free posets have recently taken some importance and motivated many studies. This class of posets introduced by Grillet [8] and Heuchenne [11] are very related to another important class of posets, namely the series-parallel posets, introduced by Lawler [12] and studied by Valdes et al. [21]. This paper shows how N-free posets can be considered as generalizations of series-parallel posets, by giving a recursive construction of N-free posets. Furthermore we propose a linear time algorithm to recognize and decompose any N-free poset. This yields some very naturel problems, namely: which are the properties(such as linear time algorithm for some invariant) of series-parallel posets that are kept for N-free posets?