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

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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 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_S(σ) 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_revS(Δ(σ τ)) m_S(Δ(Δ(σ)*Δτ for all S [n]. To prove this fact, we also prove that m_S(Δ(σ)) m_S(Δ(Δ(σ) )) for all S [n] and for all nonsingular matrices, where Δ(σ) is the simplicial complex defined by J_{Δ(σ)}= (J_{σ})).

Original languageEnglish
Pages (from-to)327-336
Number of pages10
JournalArkiv for Matematik
Volume45
Issue number2
DOIs
Publication statusPublished - 2007 Oct 1
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics(all)

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