Generic initial ideals and exterior algebraic shifting of the join of simplicial complexes

Satoshi Murai Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University

TBD mathscidoc:1701.333104

Arkiv for Matematik, 45, (2), 327-336, 2006.7
In this paper, the relation between algebraic shifting and join which was conjectured by Eran Nevo will be proved. Let σ and τ be simplicial complexes and σ*τ be their join. Let$J$_{σ}be the exterior face ideal of σ and Δ(σ) the exterior algebraic shifted complex of σ. Assume that σ*τ is a simplicial complex on [$n$]={1,2,...,$n$}. For any$d$-subset$S$⊂[$n$], let $m_{\preceq_{\textrm{rev}}S}(\sigma)$ denote the number of$d$-subsets$R$∈σ which are equal to or smaller than$S$with respect to the reverse lexicographic order. We will prove that $m_{\preceq_{\textrm{rev}}S}(\Delta(\sigma*\tau))\geq m_{\preceq_{\textup{rev}}S}(\Delta(\Delta(\sigma) *\Delta(\tau)))$ for all$S$⊂[$n$]. To prove this fact, we also prove that $m_{\preceq_{\textrm{rev}}S}(\Delta(\sigma))\geq m_{\preceq_{\textup{rev}}S}(\Delta(\Delta_{\varphi}(\sigma)))$ for all$S$⊂[$n$] and for all nonsingular matrices ϕ, where Δ_{ϕ}(σ) is the simplicial complex defined by $J_{\Delta_{\varphi}(\sigma)}=\textup{in}(\varphi(J_{\sigma}))$ .
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@inproceedings{satoshi2006generic,
  title={Generic initial ideals and exterior algebraic shifting of the join of simplicial complexes},
  author={Satoshi Murai},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203619768206913},
  booktitle={Arkiv for Matematik},
  volume={45},
  number={2},
  pages={327-336},
  year={2006},
}
Satoshi Murai. Generic initial ideals and exterior algebraic shifting of the join of simplicial complexes. 2006. Vol. 45. In Arkiv for Matematik. pp.327-336. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203619768206913.
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