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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.03500 |
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Table of Contents:
- Let $Δ$ be a $(d-1)$-dimensional simplicial complex and $h^ Δ= (h_0^ Δ,\ldots, h_d^ Δ)$ its $h$-vector. For a face uniform subdivision operation ${\mathcal F}$ we write $Δ_{\mathcal F}$ for the subdivided complex and $H_{\mathcal F}$ for the matrix such that $h^ {Δ_{\mathcal F}} = H_{\mathcal F} h^ Δ$. In connection with the real rootedness of symmetric decompositions Athanasiadis and Tzanaki studied for strictly positive $h$-vectors the inequalities $\frac{h_0^ Δ}{h_1^ Δ} \leq \frac{h_1^Δ}{h_{d-1}^ Δ} \leq \cdots \leq \frac{h_d^ Δ}{h_0^Δ}$ and $\frac{h_1^Δ}{h_{d-1}^Δ} \geq \cdots \geq \frac{h_{d-2}^Δ}{h_2^Δ} \geq \frac{h_{d-1}^Δ}{h_1^Δ}$. In this paper we show that if the inequalities holds for a simplicial complex $Δ$ and $H_{\mathcal F}$ is TP$_2$ (all entries and two minors are non-negative) then the inequalities hold for $Δ_{\mathcal F}$. 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.