Related papers: Partition and Cohen-Macaulay Extenders

Let $\Phi$ be a finite root system of rank $n$ and let $m$ be a nonnegative integer. The generalized cluster complex $\Delta^m (\Phi)$ was introduced by S. Fomin and N. Reading. It was conjectured by these authors that $\Delta^m (\Phi)$ is…

Via the BGG correspondence a simplicial complex Delta on [n] is transformed into a complex of coherent sheaves on P^n-1. We show that this complex reduces to a coherent sheaf F exactly when the Alexander dual Delta^* is Cohen-Macaulay. We…

Let $K$ be a field and $R=K[x_1,\ldots, x_n]$ be the polynomial ring in $n$ variables over a field $K$. Let $\Delta$ be a simplicial complex on $n$ vertices and $I=I_{\Delta}$ be its Stanley-Reisner ideal. In this paper, we show that if $I$…

We show that an 'almost strong Lefschetz' property holds for the barycentric subdivision of a shellable complex. From this we conclude that for the barycentric subdivision of a Cohen-Macaulay complex, the $h$-vector is unimodal, peaks in…

To a finite ranked poset $\Gamma$ we associate a finite-dimensional graded quadratic algebra $R_\Gamma$. Assuming $\Gamma$ satisfies a combinatorial condition known as uniform, $R_{\Gamma}$ is related to a well-known algebra, the splitting…

We extend the definition of chordal from graphs to clutters. The resulting family generalizes both chordal graphs and matroids, and obeys many of the same algebraic and geometric properties. Specifically, the independence complex of a…

For any flag simplicial complex $\Theta$ obtained by stellar subdividing the boundary of the cross polytope in edges, we define a flag simplicial complex $\Gamma(\Theta)$ (dependent on the sequence of subdivisions) whose $f$-vector is the…

We study the quotient complex $\Delta (B_{lm})/S_l\wr S_m$ as a means of deducing facts about the ring $k[x_1,..., x_{lm}]^{S_l\wr S_m}$. It is shown in [He] that this quotient complex is shellable when $l=2$, implying Cohen-Macaulayness of…

The $\gamma$-vector is an important enumerative invariant of a flag simplicial homology sphere. It has been conjectured by Gal that this vector is nonnegative for every such sphere $\Delta$ and by Reiner, Postnikov and Williams that it…

Ehrenborg, Govindaiah, Park, and Readdy recently introduced the van der Waerden complex, a pure simplicial complex whose facets correspond to arithmetic progressions. Using techniques from combinatorial commutative algebra, we classify when…

For a simplicial complex $\Delta$, the affect of the expansion functor on combinatorial properties of $\Delta$ and algebraic properties of its Stanley-Reisner ring has been studied in some previous papers. In this paper, we consider the…

In this paper we study the Alexander dual of a vertex decomposable simplicial complex. We define the concept of a vertex splittable ideal and show that a simplicial complex $\Delta$ is vertex decomposable if and only if $I_{\Delta^{\vee}}$…

In a recent paper, Duval, Goeckner, Klivans and Martin disproved the longstanding conjecture by Stanley, that every Cohen-Macaulay simplicial complex is partitionable. We construct counterexamples to this conjecture that are even…

For a positive integer $k$ and a non-negative integer $t$ a class of simplicial complexes, to be denoted by $k$-${\rm CM}_t$, is introduced. This class generalizes two notions for simplicial complexes: being $k$-Cohen-Macaulay and…

Given a simple undirected graph $G$ there is a simplicial complex $\mathrm{Ind}(G)$, called the independence complex, whose faces correspond to the independent sets of $G$. This is a well studied concept because it provides a fertile ground…

We introduce sequentially $S_r$ modules over a commutative graded ring and sequentially $S_r$ simplicial complexes. This generalizes two properties for modules and simplicial complexes: being sequentially Cohen-Macaulay, and satisfying…

We consider a $q$-analogue of abstract simplicial complexes, called $q$-complexes, and discuss the notion of shellability for such complexes. It is shown that $q$-complexes formed by independent subspaces of a $q$-matroid are shellable.…

Monomial ideals which are generic with respect to either their generators or irreducible components have minimal free resolutions derived from simplicial complexes. For a generic monomial ideal, the associated primes satisfy a saturated…

Starting from an unpublished conjecture of Kalai and from a conjecture of Eisenbud, Green and Harris, we study several problems relating h-vectors of Cohen-Macaulay, flag simplicial complexes and face vectors of simplicial complexes.

We disprove a long-standing open conjecture due to Simon stating that all skeleta of simplices are extendably shellable. In particular, for every $d \geq 3$ we provide a pure $d$-dimensional shellable simplicial complex which is not…