Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.03500 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915002163658752 |
|---|---|
| author | Mu, Lili Welker, Volkmar |
| author_facet | Mu, Lili Welker, Volkmar |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_03500 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Total positivity and two inequalities by Athanasiadis and Tzanaki Mu, Lili Welker, Volkmar Combinatorics 05E45, 05A20 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. |
| title | Total positivity and two inequalities by Athanasiadis and Tzanaki |
| topic | Combinatorics 05E45, 05A20 |
| url | https://arxiv.org/abs/2404.03500 |