Vertex Decomposability of Path Complexes and Stanley’s Conjectures

Seyed Mohammad Ajdani,Francisco Bulnes

doi:10.5772/intechopen.101083

Abstract

Monomials are the link between commutative algebra and combinatorics. With a simplicial complex Δ, one can associate two square-free monomial ideals: the Stanley-Reisner ideal IΔ whose generators correspond to the non-face of Δ, or the facet ideal I(Δ) that is a generalization of edge ideals of graphs and whose generators correspond to the facets of Δ. The facet ideal of a simplicial complex was first introduced by Faridi in 2002. Let G be a simple graph. The edge ideal I(G) of a graph G was first considered by R. Villarreal in 1990. He studied algebraic properties of I(G) using a combinatorial language of G. In combinatorial commutative algebra, one can attach a monomial ideal to a combinatorial object. Then, algebraic properties of this ideal are studied using combinatorial properties of combinatorial object. One of interesting problems in combinatorial commutative algebra is the Stanley’s conjectures. The Stanley’s conjectures are studied by many researchers. Let R be a Nn-graded ring and M a Zn-graded R-module. Then, Stanley conjectured that depthM≤sdepthM. He also conjectured that each Cohen-Macaulay simplicial complex is partition-able. In this chapter, we study the relation between vertex decomposability of some simplicial complexes and Stanley’s conjectures.

Highlights

Let R 1⁄4 K1⁄2x1, ... , xn, where K is a field
A sequence xi1, ... , xit of distinct vertices is called a path of length t if there are t À 1 distinct directed edges e1, ... , etÀ1 where e j is a directed edge from xi j to xi jþ1
The path ideal of G of length t is the monomial ideal ItðGÞ 1⁄4 ðxi1 ... xit : xi1, ... , xit is a path of length t in G) in the polynomial ring R 1⁄4 K1⁄2x1, ... , xn

Summary

Introduction

Let R 1⁄4 K1⁄2x1, ... , xn, where K is a field. Fix an integer n ≥ t ≥ 2 and let G be a directed graph. Xit is a path of length t in G) in the polynomial ring R 1⁄4 K1⁄2x1, ... Stanley [9] conjectured that depthðMÞ ≤ sdepthðMÞ (3) He conjectured in [10] that each Cohen-Macaulay simplicial complex is partitionable.

Preliminaries

Vertex decomposability path complexes of cycles

Vertex decomposability path complexes of trees

ΔtðGÞ is vertex decomposable

Vertex decomposability path complexes of dynkin graphs

Stanley decompositions