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 Jσ 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 $m_{\preceq_{\textrm{rev}}S}(\sigma)$ 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 $m_{\preceq_{\textrm{rev}}S}(\Delta(\sigma*\tau))\geq m_{\preceq_{\textup{rev}}S}(\Delta(\Delta(\sigma) *\Delta(\tau)))$ for all S⊂[n]. To prove this fact, we also prove that $m_{\preceq_{\textrm{rev}}S}(\Delta(\sigma))\geq m_{\preceq_{\textup{rev}}S}(\Delta(\Delta_{\varphi}(\sigma)))$ for all S⊂[n] and for all nonsingular matrices ϕ, where Δϕ(σ) is the simplicial complex defined by $J_{\Delta_{\varphi}(\sigma)}=\textup{in}(\varphi(J_{\sigma}))$.
Citation
Satoshi Murai. "Generic initial ideals and exterior algebraic shifting of the join of simplicial complexes." Ark. Mat. 45 (2) 327 - 336, October 2007. https://doi.org/10.1007/s11512-006-0030-9
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