Related papers: Lefschetz properties of balanced 3-polytopes
Generalizing the strong Lefschetz property for an $\mathbb{N}$-graded algebra, we introduce the multigraded strong Lefschetz property for an $\mathbb{N}^m$-graded algebra. We show that, for $\mathbf{a} \in \mathbb{N}^m_+$, the generic…
A simplicial complex of dimension $d-1$ is said to be balanced if its graph is $d$-colorable. Juhnke-Kubitzke and Murai proved an analogue of the generalized lower bound theorem for balanced simplicial polytopes. We establish a…
A remarkable and important property of face numbers of simplicial polytopes is the generalized lower bound inequality, which says that the $h$-numbers of any simplicial polytope are unimodal. Recently, for balanced simplicial $d$-polytopes,…
A $(d-1)$-dimensional simplicial complex $\Delta$ is balanced if its graph $G(\Delta)$ is $d$-colorable. Klee and Novik obtained the balanced lower bound theorem for balanced normal $(d-1)$-pseudomanifolds $\Delta$ with $d\geq3$ by showing…
A $(d-1)$-dimensional simplicial complex is called balanced if its underlying graph admits a proper $d$-coloring. We show that many well-known face enumeration results have natural balanced analogs (or at least conjectural analogs).…
We study simplicial complexes with a given number of vertices whose Stanley-Reisner ring has the minimal possible Betti numbers. We find that these simplicial complexes have very special combinatorial and topological structures. For…
Given any order ideal $U$ consisting of color-squarefree monomials involving variables with $d$ colors, we associate to it a balanced $(d-1)$-dimensional simplicial complex $\Delta_{\mathrm{bal}}(U)$ that we call a balanced squeezed…
In this paper, we provide a simple proof for the fact that two simplicial complexes are isomorphic if and only if their associated Stanley-Reisner rings, or their associated facet rings are isomorphic as $K$-algebras. As a consequence, we…
The van der Waerden simplicial complex, denoted ${\tt vdw}(n,k)$, is the simpicial complex whose facets correspond to the arithmetic progressions of length $k$ in the set $\{1,\ldots,n\}$. We study the Lefschetz properties of the Artinian…
Stanley proved that for any centrally symmetric simplicial $d$-polytope $P$ with $d\geq 3$, $g_2(P) \geq {d \choose 2}-d$. We provide a characterization of centrally symmetric $d$-polytopes with $d\geq 4$ that satisfy this inequality as…
An SI-sequence is a finite sequence of positive integers which is symmetric, unimodal and satisfies a certain growth condition. These are known to correspond precisely to the possible Hilbert functions of Artinian Gorenstein algebras with…
Let K be an algebraically closed field of characteristic p > 0. We apply a theorem of C. Han to give an explicit description for the weak Lefschetz property of the monomial Artinian complete intersection A = K[X,Y,Z]/(X^d,Y^d,Z^d) in terms…
This paper has two related parts. The first generalizes Hochster's formula on resolutions of Stanley-Reisner rings to a colorful version, applicable to any proper vertex-coloring of a simplicial complex. The second part examines a universal…
In this work a combinatorial approach towards the weak Lefschetz property is developed that relates this property to enumerations of signed perfect matchings as well as to enumerations of signed families of non-intersecting lattice paths in…
We investigate some combinatorial properties of convex polytopes simple in edges. For polytopes whose nonsimple vertices are located sufficiently far one from another, we prove an analog of the Hard Lefschetz theorem. It implies Stanley's…
This expository note illustrates toric perturbation and biased pairing theory to show that Artinian reductions of face rings of $2$-spheres that do not satisfy the Lefschetz property can be cut along a flat equator. This complements…
Chromatic symmetric functions are well-studied symmetric functions in algebraic combinatorics that generalize the chromatic polynomial and are related to Hessenberg varieties and diagonal harmonics. Motivated by the Stanley--Stembridge…
Stanley showed that monomial complete intersections have the strong Lefschetz property. Extending this result we show that a simple extension of an Artinian Gorenstein algebra with the strong Lefschetz property has again the strong…
The antiprism triangulation provides a natural way to subdivide a simplicial complex $\Delta$, similar to barycentric subdivision, which appeared independently in combinatorial algebraic topology and computer science. It can be defined as…
An edge-colored graph is said to be balanced if it has an equal number of edges of each color. Given a graph $G$ whose edges are colored using two colors and a positive integer $k$, the objective in the Edge Balanced Connected Subgraph…