Related papers: Cohen-Macaulay graphs and face vectors of flag com…
In this study, we consider the positive cluster complex, a full subcomplex of a cluster complex the vertices of which are all non-initial cluster variables. In particular, we provide a formula for the difference in face vectors of positive…
We present a combinatorial procedure (based on the W-graph of the Coxeter group) which shows that the characters of many intersection cohomology complexes on low rank complex flag varieties with coefficients in an arbitrary field are given…
For a graph $G$, Bolognini et al. have shown $J_{G}$ is strongly unmixed $\Rightarrow$ $J_{G}$ is Cohen-Macaulay $\Rightarrow$ $G$ is accessible, where $J_{G}$ denotes the binomial edge ideals of $G$. Accessible and strongly unmixed…
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…
A simplicial complex $\Delta$ is called flag if all minimal nonfaces of $\Delta$ have at most two elements. The following are proved: First, if $\Delta$ is a flag simplicial pseudomanifold of dimension $d-1$, then the graph of $\Delta$ (i)…
We prove that if a pure simplicial complex of dimension d with n facets has the least possible number of (d-1)-dimensional faces among all complexes with n faces of dimension d, then it is vertex decomposable. This answers a question of J.…
We prove that any rank two arithmetically Cohen-Macaulay vector bundle on a general hypersurface of degree at least three in $\mathbb{P}^5$ must be split.
Let $R=K[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over a field $K$. We show that if $G$ is a connected graph with a basic $5$-cycle $C$, then $G$ is a sequentially Cohen-Macaulay graph if and only if there exists a shedding…
We classify the Cohen-Macaulay binomial edge ideals of cactus and bicyclic graphs.
In this paper, we investigate a relation between finite graphs, simplicial flag complexes and right-angled Coxeter groups, and we provide a class of reconstructible finite graphs. We show that if $\Gamma$ is a finite graph which is the…
Given a projective algebraic set X, its dual graph G(X) is the graph whose vertices are the irreducible components of X and whose edges connect components that intersect in codimension one. Hartshorne's connectedness theorem says that if…
We describe combinatorially the Cohen-Macaulay type of edge-weighted r-path suspensions of edge-weighted graphs for an arbitrary positive integer r. The computation of the Cohen-Macaulay type of edge-weighted suspensions of edge-weighted…
We investigate when the independence complex of $G[H]$, the lexicographical product of two graphs $G$ and $H$, is either vertex decomposable or shellable. As an application, we construct an infinite family of graphs with the property that…
Consider a flag variety $Fl$ over an algebraically closed field, and a subvariety $V$ whose cycle class is a multiplicity-free sum of Schubert cycles. We show that $V$ is arithmetically normal and Cohen-Macaulay, in the projective embedding…
This paper continues investigation of the class of flag simple polytopes called 2-truncated cubes. It is an extended version of the short note Volodin (2012). A 2-truncated cube is a polytope obtained from a cube by sequence of truncations…
Monomial ideals corresponding to strong quasi-n-partite graphs are considered. Some algebraic and combinatorial properties of generalized graph ideals of a strong quasi-n-partite graph are studied. Furthermore, we show that the edge ideal…
Let X,Y be finite sets and T a set of functions from X -> Y which we will call "tableaux". We define a simplicial complex whose facets, all of the same dimension, correspond to these tableaux. Such "tableau complexes" have many nice…
A Gelfand-Cetlin polytope is a convex polytope obtained as an image of certain completely integrable system on a partial flag variety. In this paper, we give an equivalent description of the face structure of a GC-polytope in terms of so…
In recent work of Braden, Huh, Matherne, Proudfoot and Wang, a class of simplicial complexes associated to matroids, called augmented Bergman complexes, was introduced. The present article concerns the face enumeration of these complexes.…
In this paper, we introduce a topology on the set of isomorphism classes of finitely generated modules over an associative algebra. Then we focus on the relative topology on the set of isomorphism classes of maximal Cohen--Macaulay modules…