[Home ] [Archive]   [ فارسی ] Search Submit Contact  Volume 8, Issue 4 (Vol. 8,No. 4, 2022) 2022, 8(4): 164-179 Back to browse issues page
New methods for constructing shellable simplicial complexes
Mohammad Farrokhi D. G.1, Ali Akbar Yazdan Pour2
1- Institute for Advanced Studies in Basic Sciences (IASBS)
2- Institute for Advanced Studies in Basic Sciences (IASBS) , yazdan@iasbs.ac.ir
Abstract:   (179 Views)
A clutter \$mathcal{C}\$ with vertex set \$[n]\$ is an antichain of subsets of \$[n]\$, called circuits, covering all vertices. The clutter is \$d\$-uniform if all of its circuits have the same cardinality \$d\$. If \$mathbb{K}\$ is a field, then there is a one-to-one correspondence between clutters on \$V\$ and square-free monomial ideals in \$mathbb{K}[x_1,ldots,x_n]\$ as follows: To each clutter \$mathcal{C}\$ we correspond its circuit ideal \$I(mathcal{C})\$ generated by monomials \$x_{i_1}cdots x_{i_k}\$ with \${i_1,ldots,i_k}inmathcal{C}\$. Conversely, to each square-free monomial ideal \$I\$ with minimal set of generators \$mathcal{G}(I)\$, we correspond a clutter with circuits \${i_1,ldots,i_k}\$, where \$x_{i_1}cdots x_{i_k}inmathcal{G}(I)\$. The independence complex of a clutter \$mathcal{C}\$ on \$[n]\$ is the simplicial complex \$Delta_{mathcal{C}}\$ whose faces are independent sets in \$mathcal{C}\$ by which we mean sets \$Fsubseteq [n]\$ such that \$ensubseteq F\$ for all \$einmathcal{C}\$. It is easy to see that the Stanley-Reisner ideal of \$Delta_{mathcal{C}}\$ coincides with \$I(mathcal{C})\$. The above correspondence establishes a one-to-one correspondence between simplicial complexes and independence complex of clutters. A simplicial complex \$Delta\$ is shellable if there exists a total order on its facets, say \$F_1

Keywords: Clutter, Hybrid clutter, Shellability, Cohen-Macaulay, Independence complex
Type of Study: S | Subject: alg
Received: 2021/02/15 | Revised: 2023/01/25 | Accepted: 2021/05/1 | Published: 2022/12/31 | ePublished: 2022/12/31
Send email to the article author        Rights and permissions This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
 Rights and permissions This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.  