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

Abstract

In this paper, the relation between algebraic shifting and join which was conjectured by Eran Nevo will be proved. Let $\sigma$ and $\tau$ be simplicial complexes and $\sigma * \tau$ their join. Let $J_\sigma$ be the exterior face ideal of $\sigma$ and $\Delta(\sigma)$ the exterior algebraic shifted complex of $\sigma$. Assume that $\sigma * \tau$ is a simplicial complex on $[n]=\{1,2,...,n\}$. For any $d$-subset $S \subset [n]$, let $m_{\preceq_{rev} S}(\sigma)$ denote the number of $d$-subsets $R \in \sigma$ which is equal to or smaller than $S$ w.r.t. the reverse lexicographic order. We will prove that $m_{\preceq_{rev} S}(\Delta({\sigma * \tau}))\geq m_{\preceq_{rev} S}(\Delta ({\Delta(\sigma)} * {\Delta (\tau)}))$ for all $S \subset [n]$. To prove this fact, we also prove that $m_{\preceq_{rev} S}(\Delta(\sigma))\geq m_{\preceq_{rev} S}(\Delta({\Delta_{\phi}(\sigma)}))$ for all $S\subset [n]$ and for all non-singular matrices $\phi$, where $\Delta_{\phi}(\sigma)$ is the simplicial complex defined by $J_{\Delta_{\phi}(\sigma)}=\init(\phi(J_\sigma))$.