bibkey: delascuevasetal2024invariantpolynomialdecompositions
authors: Gemma De las Cuevas; Andreas Klingler; Tim Netzer
year: 2024
title: "Polynomial decompositions with invariance and positivity inspired by tensors"
doi: 10.1016/j.laa.2024.05.025
url: https://arxiv.org/abs/2109.06680v2
claim: "The paper defines invariant decompositions of polynomials over weighted simplicial complexes with group actions, introduces factorizable (Omega,G) structures through a positive equivariant solution of a product system with right-hand side K_alpha^{-1}, uses factorizability as a hypothesis of its rank inequalities, shows that a free vertex action and the double edge with copy-swapping action are factorizable, and asks whether non-factorizable (Omega,G) structures exist."
strata_touched:
  - D5/S3/Quantum/TensorNetworks/NonFactorizableDoubledTriangle
license: citation-only
triage: anchor

Polynomial decompositions with invariance and positivity inspired by tensors

G. De las Cuevas, A. Klingler and T. Netzer, arXiv:2109.06680v2; Linear Algebra and its Applications 698 (2024), 537–588. Quotations are from the v2 TeX source; displayed macros are expanded without changing the wording.

Weighted simplicial complexes (Definition def:wsc):

A weighted simplicial complex on $[n]$ is a map $\Omega :{\mathcal{P}}_n\to \mathbb{N}$ such that $\Omega (S_1)$ divides $\Omega (S_2)$ whenever $S_1\subseteq S_2$.

A set $S\in {\mathcal{P}}_n$ is called a simplex of $\Omega$ if $\Omega (S)\ne 0$. We will always assume that each singleton $\{i\}$ is a simplex, and call the elements $i\in [n]$ the vertices of the weighted simplicial complex. We call a maximal simplex (with respect to inclusion) a facet of $\Omega .$

$\tilde{\mathcal{F}}$ contains each facet $F$ exactly $\Omega (F)$-many times.

A group action on $\Omega$ consists of an action on $[n]$ leaving $\Omega$ invariant and

An action of $G$ on the set of multifacets $\tilde{\mathcal{F}}$ such that the canonical collapse map $c:\tilde{\mathcal{F}}\to \mathcal{F}$ is $G$-linear.

Factorizability (Definition before Example “Factorizable group actions”):

We say that $(\Omega ,G)$ is factorizable if for each finite index set $\mathcal{I}$ the following system of equations admits a solution with all $C_{\beta }^{[i]}>0$ and $C_{{}^g\beta }^{[gi]}=C_{\beta }^{[i]}$ for all $i\in [n]$, $\beta \in {\mathcal{I}}^{{\tilde{\mathcal{F}}}_i}$, and $g\in G$: $$C_{{\alpha }_{{|}_0}}^{[0]}\cdot C_{{\alpha }_{{|}_1}}^{[1]}\cdots C_{{\alpha }_{{|}_n}}^{[n]}=K_{\alpha }^{-1}\text{ for all }\alpha \in {\mathcal{I}}^{\tilde{\mathcal{F}}},$$ where $$K_{\alpha }:=|\{\gamma \in {\mathcal{I}}^{\tilde{\mathcal{F}}}:\exists g_0,\dots ,g_n\in G\text{ with }{g}_{i}i=i\text{ and }{\left({}^{g_i}\gamma \right)}_{{|}_i}={\alpha }_{{|}_i}\text{ for all }i\in [n]\}|.$$

The open question (Question 38, PDF p. 28 of v2):

In fact, we are not aware of any non-factorizable $(\Omega ,G)$ structures, leading to the following open question.

Are there non-factorizable $(\Omega ,G)$ structures?

The conclusion repeats it as one of two “immediate” open questions. Proposition 39(iii) assumes factorizability for ${sos-rank}_{(\Omega ,G)}\le {sep-rank}_{(\Omega ,G)}$; Proposition 39(iv) assumes a free action on a connected $\Omega$ together with factorizability.

Verified locator

Source: https://arxiv.org/abs/2109.06680v2

• URL: https://arxiv.org/abs/2109.06680v2 (the e-print is a single gzip file whose header names Manuscript_final.tex; md5 e601e0a3202f352cba4bf6d4d13e7db4, 2167 lines):
  • Definition def:wsc, l. 318–341;
  • Definition def:action and the group action on a weighted simplicial complex, l. 395–440;
  • Definition (Factorizable), l. 1515–1522;
  • Example (Factorizable group actions), l. 1528–1542;
  • Question qu:fact, l. 1544–1549;
  • Proposition pro:inequalitites, l. 1553–1566;
  • the conclusion’s restatement, l. 2034–2035.
• DOI: 10.1016/j.laa.2024.05.025 (https://doi.org/10.1016/j.laa.2024.05.025; Crossref lists Linear Algebra and its Applications, volume 698, pages 537–588, October 2024).