scholarly journals Algebraic shifting and strongly edge decomposable complexes

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Satoshi Murai
Keyword(s):  
Simplicial Complex ◽  
Boundary Complex ◽  
International Audience ◽  
Simplicial Sphere ◽  
Algebraic Shifting

International audience Let $\Gamma$ be a simplicial complex with $n$ vertices, and let $\Delta (\Gamma)$ be either its exterior algebraic shifted complex or its symmetric algebraic shifted complex. If $\Gamma$ is a simplicial sphere, then it is known that (a) $\Delta (\Gamma)$ is pure and (b) $h$-vector of $\Gamma$ is symmetric. Kalai and Sarkaria conjectured that if $\Gamma$ is a simplicial sphere then its algebraic shifting also satisfies (c) $\Delta (\Gamma) \subset \Delta (C(n,d))$, where $C(n,d)$ is the boundary complex of the cyclic $d$-polytope with $n$ vertices. We show this conjecture for strongly edge decomposable spheres introduced by Nevo. We also show that any shifted simplicial complex satisfying (a), (b) and (c) is the algebraic shifted complex of some simplicial sphere.

Algebraic Shifting and Sequentially Cohen-Macaulay Simplicial Complexes

10.37236/1245 ◽  
1996 ◽  
Vol 3 (1) ◽  
Author(s):  
Art M. Duval
Keyword(s):  
Simplicial Complex ◽  
Simplicial Complexes ◽  
Pure Case ◽  
Definition Of ◽  
Algebraic Shifting

Björner and Wachs generalized the definition of shellability by dropping the assumption of purity; they also introduced the $h$-triangle, a doubly-indexed generalization of the $h$-vector which is combinatorially significant for nonpure shellable complexes. Stanley subsequently defined a nonpure simplicial complex to be sequentially Cohen-Macaulay if it satisfies algebraic conditions that generalize the Cohen-Macaulay conditions for pure complexes, so that a nonpure shellable complex is sequentially Cohen-Macaulay. We show that algebraic shifting preserves the $h$-triangle of a simplicial complex $K$ if and only if $K$ is sequentially Cohen-Macaulay. This generalizes a result of Kalai's for the pure case. Immediate consequences include that nonpure shellable complexes and sequentially Cohen-Macaulay complexes have the same set of possible $h$-triangles.

scholarly journals Enumerating Triangulations of Convex Polytopes

2001 ◽  
Vol DMTCS Proceedings vol. AA,... (Proceedings) ◽  
Author(s):  
Sergei Bespamyatnikh
Keyword(s):  
Convex Hull ◽  
Simplicial Complex ◽  
Convex Polytopes ◽  
Three Dimensions ◽  
Finite Point ◽  
Convex Position ◽  
International Audience ◽  
Point Set

International audience A triangulation of a finite point set A in $\mathbb{R}^d$ is a geometric simplicial complex which covers the convex hull of $A$ and whose vertices are points of $A$. We study the graph of triangulations whose vertices represent the triangulations and whose edges represent geometric bistellar flips. The main result of this paper is that the graph of triangulations in three dimensions is connected when the points of $A$ are in convex position. We introduce a tree of triangulations and present an algorithm for enumerating triangulations in $O(log log n)$ time per triangulation.

scholarly journals Critical Groups of Simplicial Complexes

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Art M. Duval ◽  
Caroline J. Klivans ◽  
Jeremy L. Martin
Keyword(s):  
Simplicial Complex ◽  
Spanning Trees ◽  
Simplicial Complexes ◽  
Chow Groups ◽  
Critical Group ◽  
Critical Groups ◽  
Combinatorial Laplacian ◽  
Nous Montrons ◽  
International Audience ◽  
Laplacian Operators

International audience We generalize the theory of critical groups from graphs to simplicial complexes. Specifically, given a simplicial complex, we define a family of abelian groups in terms of combinatorial Laplacian operators, generalizing the construction of the critical group of a graph. We show how to realize these critical groups explicitly as cokernels of reduced Laplacians, and prove that they are finite, with orders given by weighted enumerators of simplicial spanning trees. We describe how the critical groups of a complex represent flow along its faces, and sketch another potential interpretation as analogues of Chow groups. Nous généralisons la théorie des groupes critiques des graphes aux complexes simpliciaux. Plus précisément, pour un complexe simplicial, nous définissons une famille de groupes abéliens en termes d'opérateurs de Laplace combinatoires, qui généralise la construction du groupe critique d'un graphe. Nous montrons comment réaliser ces groupes critiques explicitement comme conoyaux des opérateurs de Laplace réduits combinatoires, et montrons qu'ils sont finis. Leurs ordres sont obtenus en comptant (avec des poids) des arbres simpliciaux couvrants. Nous décrivons comment les groupes critiques d'un complexe représentent le flux le long de ses faces, et esquissons une autre interprétation potentielle comme analogues des groupes de Chow.

scholarly journals Coherent fans in the space of flows in framed graphs

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Vladimir I. Danilov ◽  
Alexander V. Karzanov ◽  
Gleb A. Koshevoy
Keyword(s):  
Simplicial Complex ◽  
Binary Relation ◽  
Directed Graph ◽  
The Other ◽  
Cluster Algebras ◽  
Linear Orders ◽  
International Audience ◽  
Acyclic Directed Graph ◽  
Extreme Rays ◽  
Space Of Flows

International audience Let $G=(V,E)$ be a finite acyclic directed graph. Being motivated by a study of certain aspects of cluster algebras, we are interested in a class of triangulations of the cone of non-negative flows in $G, \mathcal F_+(G)$. To construct a triangulation, we fix a raming at each inner vertex $v$ of $G$, which consists of two linear orders: one on the set of incoming edges, and the other on the set of outgoing edges of $v$. A digraph $G$ endowed with a framing at each inner vertex is called $framed$. Given a framing on $G$, we define a reflexive and symmetric binary relation on the set of extreme rays of $\mathcal F_+ (G)$. We prove that that the complex of cliques formed by this binary relation is a pure simplicial complex, and that the cones spanned by cliques constitute a unimodular simplicial regular fan $Σ (G)$ covering the entire $\mathcal F_+(G)$. Soit $G=(V,E)$ un graphe orientè, fini et acyclique. Nous nous intéressons, en lien avec l’étude de certains aspects des algèbres amassées, à une classe de triangulations du cône des flots positifs de $G, \mathcal F_+(G)$. Pour construire une triangulation, nous ajoutons une structure en chaque sommet interne $v$ de $G$, constituée de deux ordres totaux : l'un sur l'ensemble des arcs entrants, l'autre sur l'ensemble des arcs sortants de $v$. On dit alors que $G$ est structurè. On définit ensuite une relation binaire réflexive et symétrique sur l'ensemble des rayons extrêmes de $\mathcal F_+ (G)$. Nous démontrons que le complexe des cliques formè par cette relation binaire est un complexe simplicial pur, et que le cône engendré par les cliques forme un éventail régulier simplicial unimodulaire $Σ (G)$ qui couvre complètement $\mathcal F_+(G)$.

scholarly journals Rational Catalan Combinatorics: The Associahedron

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Drew Armstrong ◽  
Brendon Rhoades ◽  
Nathan Williams
Keyword(s):  
Simplicial Complex ◽  
Rational Number ◽  
Catalan Number ◽  
Positive Rational Number ◽  
Dyck Paths ◽  
Narayana Numbers ◽  
International Audience ◽  
Positive Integers ◽  
Coprime Positive Integers ◽  
Product Formulas

International audience Each positive rational number $x>0$ can be written $\textbf{uniquely}$ as $x=a/(b-a)$ for coprime positive integers 0<$a$<$b$. We will identify $x$ with the pair $(a,b)$. In this extended abstract we use $\textit{rational Dyck paths}$ to define for each positive rational $x>0$ a simplicial complex $\mathsf{Ass} (x)=\mathsf{Ass} (a,b)$ called the $\textit{rational associahedron}$. It is a pure simplicial complex of dimension $a-2$, and its maximal faces are counted by the $\textit{rational Catalan number}$ $\mathsf{Cat} (x)=\mathsf{Cat}(a,b):=\frac{(a+b-1)! }{ a! b!}.$ The cases $(a,b)=(n,n+1)$ and $(a,b)=(n,kn+1)$ recover the classical associahedron and its Fuss-Catalan generalization studied by Athanasiadis-Tzanaki and Fomin-Reading. We prove that $\mathsf{Ass} (a,b)$ is shellable and give nice product formulas for its $h$-vector (the $\textit{rational Narayana numbers}$) and $f$-vector (the $\textit{rational Kirkman numbers}$). We define $\mathsf{Ass} (a,b)$ .

scholarly journals Wedge Operations and Torus Symmetries II

2017 ◽  
Vol 69 (4) ◽  
pp. 767-789 ◽  
Author(s):  
Suyoung Choi ◽  
Hanchul Park
Keyword(s):  
Positive Integer ◽  
Simplicial Complex ◽  
Main Idea ◽  
Picard Number ◽  
Fundamental Idea ◽  
Combinatorial Objects ◽  
Toric Topology ◽  
Simplicial Sphere ◽  
Fixed Positive Integer

AbstractA fundamental idea in toric topology is that classes of manifolds with well-behaved torus actions (simply, toric spaces) are classified by pairs of simplicial complexes and (non-singular) characteristic maps. In a previous paper, the authors provided a new way to find all characteristic maps on a simplicial complex K(J) obtainable by a sequence of wedgings from K.The main idea was that characteristic maps on K theoretically determine all possible characteristic maps on a wedge of K.We further develop our previous work for classification of toric spaces. For a star-shaped simplicial sphere K of dimension n-1 with m vertices, the Picard number Pic(K) of K is m-n. We call K a seed if K cannot be obtained by wedgings. First, we show that for a fixed positive integer 𝓁, there are at most finitely many seeds of Picard 𝓁 number supporting characteristic maps. As a corollary, the conjecture proposed by V. V. Batyrev in is solved affirmatively.Secondly, we investigate a systematicmethod to find all characteristic maps on K(J) using combinatorial objects called (realizable) puzzles that only depend on a seed K. These two facts lead to a practical way to classify the toric spaces of fixed Picard number.

scholarly journals Bucshbaum simplicial posets

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Jonathan Browder ◽  
Steven Klee
Keyword(s):  
Simplicial Complex ◽  
Necessary Conditions ◽  
Betti Numbers ◽  
Geometric Realization ◽  
The Family ◽  
International Audience ◽  
Simplicial Poset ◽  
Simplicial Cell ◽  
Topological Data

International audience The family of Buchsbaum simplicial posets generalizes the family of simplicial cell manifolds. The $h'-$vector of a simplicial complex or simplicial poset encodes the combinatorial and topological data of its face numbers and the reduced Betti numbers of its geometric realization. Novik and Swartz showed that the $h'-$vector of a Buchsbaum simplicial poset satisfies certain simple inequalities. In this paper we show that these necessary conditions are in fact sufficient to characterize the h'-vectors of Buchsbaum simplicial posets with prescribed Betti numbers.

scholarly journals A non-partitionable Cohen–Macaulay simplicial complex

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Art M. Duval ◽  
Bennet Goeckner ◽  
Caroline J. Klivans ◽  
Jeremy Martin
Keyword(s):  
Simplicial Complex ◽  
Monomial Ideal ◽  
Stanley Depth ◽  
International Audience

International audience A long-standing conjecture of Stanley states that every Cohen–Macaulay simplicial complex is partition- able. 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.

scholarly journals Generalized triangulations, pipe dreams, and simplicial spheres

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Luis Serrano ◽  
Christian Stump
Keyword(s):  
Simplicial Complex ◽  
Canonical Connection ◽  
Cyclic Sieving Phenomenon ◽  
North East ◽  
Schubert Polynomials ◽  
Dyck Paths ◽  
Nous Montrons ◽  
International Audience ◽  
Vertex Decomposable ◽  
Cyclic Sieving

International audience We exhibit a canonical connection between maximal $(0,1)$-fillings of a moon polyomino avoiding north-east chains of a given length and reduced pipe dreams of a certain permutation. Following this approach we show that the simplicial complex of such maximal fillings is a vertex-decomposable and thus a shellable sphere. In particular, this implies a positivity result for Schubert polynomials. For Ferrers shapes, we moreover construct a bijection to maximal fillings avoiding south-east chains of the same length which specializes to a bijection between $k$-triangulations of the $n$-gon and $k$-fans of Dyck paths. Using this, we translate a conjectured cyclic sieving phenomenon for $k$-triangulations with rotation to $k$-flagged tableaux with promotion. Nous décrivons un lien canonique entre les $(0,1)$-remplissages maximaux d'un polyomino-lune évitant les chaînes Nord-Est d'une longueur donnée, et les "pipe dreams'' réduits d'une certaine permutation. En suivant cette approche nous montrons que le complexe simplicial de tels remplissages maximaux est une sphère "vertex-decomposable'' et donc "shellable''. En particulier, cela entraîne un résultat de positivité sur les polynômes de Schubert. De plus, nous construisons, dans le cas des diagrammes de Ferrers, une bijection vers les remplissages maximaux évitant les chaînes Sud-Est de même longueur, qui se spécialise en une bijection entre les $k$-triangulations d'un $n$-gone et les $k$-faisceaux de chemins de Dyck. A l'aide de celle-ci, nous traduisons une instance conjecturale du phénomène de tamis cyclique pour les $k$-triangulations avec rotation dans le cadre des tableaux $k$-marqués avec promotion.

scholarly journals A Combinatorial Proof of a Formula for Betti Numbers of a Stacked Polytope

10.37236/281 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Suyoung Choi ◽  
Jang Soo Kim
Keyword(s):  
Simplicial Complex ◽  
Betti Number ◽  
Betti Numbers ◽  
Combinatorial Proof ◽  
Boundary Complex ◽  
Combinatorial Interpretation ◽  
Stacked Polytope

For a simplicial complex $\Delta$, the graded Betti number $\beta_{i,j}({\bf k}[\Delta])$ of the Stanley-Reisner ring ${\bf k}[\Delta]$ over a field ${\bf k}$ has a combinatorial interpretation due to Hochster. Terai and Hibi showed that if $\Delta$ is the boundary complex of a $d$-dimensional stacked polytope with $n$ vertices for $d\geq3$, then $\beta_{k-1,k}({\bf k}[\Delta])=(k-1){n-d\choose k}$. We prove this combinatorially.

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