Decompositions of Simplicial Complexes

In this thesis we study the interplay between various combinatorial, algebraic, and topological properties of simplicial complexes. We focus on when these properties imply the existence of decompositions of the face poset. In Chapter 2, we present a counterexample to Stanley's partitionability conjecture, we give a characterization of the h-vectors of Cohen-Macaulay relative complexes, and we construct a family of disconnected partitionable complexes. In Chapter 3, we introduce colorated cohomology, which aims to combine the theories of color shifting and iterated homology. Colorated cohomology gives rise to certain decompositions of balanced complexes that preserve the balanced structure. We give conditions that would guarantee the existence of a weaker form of Stanley's partitionability conjecture for balanced Cohen-Macaulay complexes. We consider Stanley's conjecture on k-fold acyclic complexes in Chapter 4, and we show that a relaxation of this conjecture holds in general. We also show that the conjecture holds in the case when k is the dimension of a given complex, and we present a framework that may lead to a counterexample to the original version of this conjecture

A non-partitionable Cohen-Macaulay simplicial complex

A long-standing conjecture of Stanley states that every Cohen-Macaulay simplicial complex is partitionable. We disprove the conjecture by constructing an explicit counterexample. Due to a result of Herzog, Jahan and Yassemi, our construction also disproves the conjecture that the Stanley depth of a monomial ideal is always at least its depth.Comment: Final version. 13 pages, 2 figure

Higher Nerves of Simplicial Complexes

We investigate generalized notions of the nerve complex for the facets of a simplicial complex. We show that the homologies of these higher nerve complexes determine the depth of the Stanley-Reisner ring $k[\Delta]$ as well as the $f$-vector and $h$-vector of $\Delta$. We present, as an application, a formula for computing regularity of monomial ideals.Comment: We rewrite Section 4 to fix some errors and clarify the proof

Lattice polytopes from Schur and symmetric Grothendieck polynomials

Given a family of lattice polytopes, two common questions in Ehrhart Theory are determining when a polytope has the integer decomposition property and determining when a polytope is reflexive. While these properties are of independent interest, the confluence of these properties is a source of active investigation due to conjectures regarding the unimodality of the $h^\ast$-polynomial. In this paper, we consider the Newton polytopes arising from two families of polynomials in algebraic combinatorics: Schur polynomials and inflated symmetric Grothendieck polynomials. In both cases, we prove that these polytopes have the integer decomposition property by using the fact that both families of polynomials have saturated Newton polytope. Furthermore, in both cases, we provide a complete characterization of when these polytopes are reflexive. We conclude with some explicit formulas and unimodality implications of the $h^\ast$-vector in the case of Schur polynomials.Comment: 37 pages, 3 tables, 4 figures; Comments Welcome; Version 2: updated references to acknowledge one result was previously known, corrected values in Table 1 and reference correct OEIS sequence; Version 3: Final Version. To appear in Electronic Journal of Combinatoric

Resolving Stanley's conjecture on kk-fold acyclic complexes

In 1993 Stanley showed that if a simplicial complex is acyclic over some field, then its face poset can be decomposed into disjoint rank $1$ boolean intervals whose minimal faces together form a subcomplex. Stanley further conjectured that complexes with a higher notion of acyclicity could be decomposed in a similar way using boolean intervals of higher rank. We provide an explicit counterexample to this conjecture. We also prove a version of the conjecture for boolean trees and show that the original conjecture holds when this notion of acyclicity is as high as possible.Mathematics Subject Classifications: 05E45, 55U1

A characterization of two-dimensional Buchsbaum matching complexes

The matching complex $M(G)$ of a graph $G$ is the set of all matchings in $G$. A Buchsbaum simplicial complex is a generalization of both a homology manifold and a Cohen--Macaulay complex. We give a complete characterization of the graphs $G$ for which $M(G)$ is a two-dimensional Buchsbaum complex. As an intermediate step, we determine which graphs have matching complexes that are themselves connected graphs.Comment: 22 pages. Minor changes throughout and a few results reordered for clarity of presentation. Some of the Buchsbaum families have been renamed to match the order in which they appear in the paper. Submitted for publicatio