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

Satoshi Murai

Published 2005-06-15, updated 2006-12-14Version 2

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 $J_{\Delta_{\phi}(\sigma)}=\init(\phi(J_\sigma))$.