https://hal.inria.fr/hal-01185188Murai, SatoshiSatoshiMuraiIST Osaka - Department of Pure and Applied Mathematics - Graduate School of Information Science and TechnologyAlgebraic shifting and strongly edge decomposable complexesHAL CCSD2008Algebraic shiftingSimplicial spheresThe strong Lefschetz propertySqueezed spheres[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO][INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]Episciences Iam, CoordinationKrattenthaler, Christian and Sagan, Bruce2015-08-19 11:45:192017-05-11 01:02:522015-08-24 10:04:06enConference papershttps://hal.inria.fr/hal-01185188/document10.46298/dmtcs.3652application/pdf1Let $\Gamma$ be a simplicial complex with $n$ vertices, and let $\Delta (\Gamma)$ be either its exterior algebraic shifted complex or its symmetric algebraic shifted complex. If $\Gamma$ is a simplicial sphere, then it is known that (a) $\Delta (\Gamma)$ is pure and (b) $h$-vector of $\Gamma$ is symmetric. Kalai and Sarkaria conjectured that if $\Gamma$ is a simplicial sphere then its algebraic shifting also satisfies (c) $\Delta (\Gamma) \subset \Delta (C(n,d))$, 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.