### Abstract

Let Γ be a simplicial complex with n vertices, and let Δ(Γ) be either its exterior algebraic shifted complex or its symmetric algebraic shifted complex. If Γ is a simplicial sphere, then it is known that (a) Δ(Γ) is pure and (b) h-vector of Γ is symmetric. Kalai and Sarkaria conjectured that if Γ is a simplicial sphere then its algebraic shifting also satisfies (c) Δ (Γ) ⊂ Δ (C(n; d)), where C(n; d) is the boundary complex of the cyclic d-polytope with n vertices. We show this conjecture for strongly edge decomposable spheres introduced by Nevo. We also show that any shifted simplicial complex satisfying (a), (b) and (c) is the algebraic shifted complex of some simplicial sphere.

Original language | English |
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Pages | 1-12 |

Number of pages | 12 |

Publication status | Published - 2008 Dec 1 |

Externally published | Yes |

Event | 20th International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC'08 - Valparaiso, Chile Duration: 2008 Jun 23 → 2008 Jun 27 |

### Other

Other | 20th International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC'08 |
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Country | Chile |

City | Valparaiso |

Period | 08/6/23 → 08/6/27 |

### Fingerprint

### Keywords

- Algebraic shifting
- Simplicial spheres
- Squeezed spheres
- The strong Lefschetz property

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Algebraic shifting and strongly edge decomposable complexes*. 1-12. Paper presented at 20th International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC'08, Valparaiso, Chile.

**Algebraic shifting and strongly edge decomposable complexes.** / Murai, Satoshi.

Research output: Contribution to conference › Paper

}

TY - CONF

T1 - Algebraic shifting and strongly edge decomposable complexes

AU - Murai, Satoshi

PY - 2008/12/1

Y1 - 2008/12/1

N2 - Let Γ be a simplicial complex with n vertices, and let Δ(Γ) be either its exterior algebraic shifted complex or its symmetric algebraic shifted complex. If Γ is a simplicial sphere, then it is known that (a) Δ(Γ) is pure and (b) h-vector of Γ is symmetric. Kalai and Sarkaria conjectured that if Γ is a simplicial sphere then its algebraic shifting also satisfies (c) Δ (Γ) ⊂ Δ (C(n; d)), where C(n; d) is the boundary complex of the cyclic d-polytope with n vertices. We show this conjecture for strongly edge decomposable spheres introduced by Nevo. We also show that any shifted simplicial complex satisfying (a), (b) and (c) is the algebraic shifted complex of some simplicial sphere.

AB - Let Γ be a simplicial complex with n vertices, and let Δ(Γ) be either its exterior algebraic shifted complex or its symmetric algebraic shifted complex. If Γ is a simplicial sphere, then it is known that (a) Δ(Γ) is pure and (b) h-vector of Γ is symmetric. Kalai and Sarkaria conjectured that if Γ is a simplicial sphere then its algebraic shifting also satisfies (c) Δ (Γ) ⊂ Δ (C(n; d)), where C(n; d) is the boundary complex of the cyclic d-polytope with n vertices. We show this conjecture for strongly edge decomposable spheres introduced by Nevo. We also show that any shifted simplicial complex satisfying (a), (b) and (c) is the algebraic shifted complex of some simplicial sphere.

KW - Algebraic shifting

KW - Simplicial spheres

KW - Squeezed spheres

KW - The strong Lefschetz property

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M3 - Paper

AN - SCOPUS:84860492709

SP - 1

EP - 12

ER -