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 |
|---|---|
| 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.
In: Arkiv for Matematik, Vol. 45, No. 2, 01.10.2007, p. 327-336.Research output: Contribution to journal › Article
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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})).
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UR - http://www.scopus.com/inward/citedby.url?scp=38149110516&partnerID=8YFLogxK
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 -