Let $\Delta$ be a $(d-1)$-dimensional simplicial complex and $h^ \Delta = (h_0^ \Delta ,\ldots, h_d^ \Delta)$ its $h$-vector. For a face uniform subdivision operation ${ \mathcal F}$ we write $\Delta_{ \mathcal F}$ for the subdivided complex and $H_{ \mathcal F}$ for the matrix such that $h^ { \Delta_{ \mathcal F}} = H_{ \mathcal F} h^ \Delta$. In connection with the real rootedness of symmetric decompositions Athanasiadis and Tzanaki studied for strictly positive $h$-vectors the inequalities $\frac{ h_0^ \Delta}{ h_1^ \Delta} \leq \frac{ h_1^\Delta}{ h_{ d-1}^ \Delta} \leq \cdots \leq \frac{ h_d^ \Delta}{ h_0^\Delta}$ and $\frac{ h_1^\Delta}{ h_{ d-1}^\Delta} \geq \cdots \geq \frac{ h_{ d-2}^\Delta}{ h_2^\Delta} \geq \frac{ h_{ d-1}^\Delta}{ h_1^\Delta}$. In this paper we show that if the inequalities holds for a simplicial complex $\Delta$ and $H_{ \mathcal F}$ is TP$_2$ (all entries and two minors are non-negative) then the inequalities hold for $\Delta_{ \mathcal F}$. We prove that if ${ \mathcal F}$ is the barycentric subdivision then $H_{ \mathcal F}$ is TP$_2$. If ${ \mathcal F}$ is the $r$\textsuperscript{ th}-edgewise subdivision then work of Diaconis and Fulman shows $H_{ \mathcal F}$ is TP$_2$. Indeed in this case by work of Mao and Wang $H_{ \mathcal F}$ is even TP.
Categories: math.CO