Total positivity and two inequalities by Athanasiadis and Tzanaki

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.