### 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 language | English |
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Pages (from-to) | 327-336 |

Number of pages | 10 |

Journal | Arkiv for Matematik |

Volume | 45 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2007 Oct 1 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Generic initial ideals and exterior algebraic shifting of the join of simplicial complexes.** / Murai, Satoshi.

Research output: Contribution to journal › Article

*Arkiv for Matematik*, vol. 45, no. 2, pp. 327-336. https://doi.org/10.1007/s11512-006-0030-9

}

TY - JOUR

T1 - Generic initial ideals and exterior algebraic shifting of the join of simplicial complexes

AU - Murai, Satoshi

PY - 2007/10/1

Y1 - 2007/10/1

N2 - 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})).

AB - 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})).

UR - http://www.scopus.com/inward/record.url?scp=38149110516&partnerID=8YFLogxK

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U2 - 10.1007/s11512-006-0030-9

DO - 10.1007/s11512-006-0030-9

M3 - Article

VL - 45

SP - 327

EP - 336

JO - Arkiv for Matematik

JF - Arkiv for Matematik

SN - 0004-2080

IS - 2

ER -