Total positivity and two inequalities by Athanasiadis and Tzanaki

Abstract

Let $\Delta$ be a $(d-1)$-dimensional simplicial complex and $h^{\Delta }=(h_0^{\Delta },\dots ,h_d^{\Delta })$ 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 $\frac{h_0^{\Delta }}{h_1^{\Delta }}\le \frac{h_1^{\Delta }}{h_{d-1}^{\Delta }}\le \cdots \le \frac{h_d^{\Delta }}{h_0^{\Delta }}$ and $\frac{h_1^{\Delta }}{h_{d-1}^{\Delta }}\ge \cdots \ge \frac{h_{d-2}^{\Delta }}{h_2^{\Delta }}\ge \frac{h_{d-1}^{\Delta }}{h_1^{\Delta }}$. 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.

Annals of Combinatorics to appear