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

Authors:	Satoshi Murai

Institution:	(1) Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka Osaka, 560-0043, Japan

Abstract:	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 στ 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 στ 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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