# MathSciDoc: An Archive for Mathematician ∫

#### TBDmathscidoc: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 $J_{\Delta_{\varphi}(\sigma)}=\textup{in}(\varphi(J_{\sigma}))$ .
@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.