Stellahedral geometry of partially ordered sets
Abstract
We introduce a transformation on partially ordered sets, termed the \emph{stellahedral transform}, with notable features. It preserves the properties of being Eulerian, Cohen--Macaulay, and of being the face poset of a polytope. Furthermore, it admits an explicit geometric realization for convex polytopes and specializes to the construction that takes a simplex to the stellahedron. One motivation for this definition comes from the theory of toric h-polynomials and (augmented) Chow polynomials of Eulerian posets. We show that the right augmented Chow polynomial of an Eulerian poset P agrees with the toric h-polynomial of the stellahedral transform of P. We use this perspective, together with cd-index results due to Ehrenborg (2005) and Karu (2006), to prove two positivity results for augmented Chow polynomials: for Gorenstein* posets they are unimodal, and for face posets of polytopes they are γ-positive. Along the way we provide negative answers to two open questions concerning Eulerian and Gorenstein* posets. First, the question on the nonnegativity of Eulerian Chow polynomials, posed by Ferroni, Matherne, and Vecchi (2024). Second, the question posed by Athanasiadis and Kalampogia-Evangelinou (2023) on the real-rootedness of chain and Chow polynomials of Gorenstein* posets: these examples provide a novel application of a technique introduced by Murai and Nevo (2014).
Related
- Homology and Topology of Partially Ordered Fields
- Does the Endomorphism Ordered Set of a Finite Ordered Set Determine the Ordered Set? The Cases of Height $1$ and "Trebled" Ordered Sets
- On stable semilinear sets in ordered vector spaces
- The AA problem for finite ordered sets
- Quasi-overlap and quasi-grouping functions on bounded pseudo-ordered sets
BibTeX
@misc{faustini2026stellahedral,
title = {{Stellahedral geometry of partially ordered sets}},
author = {Tommaso Faustini and Luis Ferroni and Ludovico Piazza},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22078},
eprint = {2609.22078},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22078},
}
RIS
TY - UNPB
TI - Stellahedral geometry of partially ordered sets
AU - Tommaso Faustini
AU - Luis Ferroni
AU - Ludovico Piazza
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.22078
UR - https://arxiv.org/abs/2609.22078
AB - We introduce a transformation on partially ordered sets, termed the \emph{stellahedral transform}, with notable features. It preserves the properties of being Eulerian, Cohen--Macaulay, and of being the face poset of a polytope. Furthermore, it admits an explicit geometric realization for convex polytopes and specializes to the construction that takes a simplex to the stellahedron. One motivation for this definition comes from the theory of toric h-polynomials and (augmented) Chow polynomials of Eulerian posets. We show that the right augmented Chow polynomial of an Eulerian poset P agrees with the toric h-polynomial of the stellahedral transform of P. We use this perspective, together with cd-index results due to Ehrenborg (2005) and Karu (2006), to prove two positivity results for augmented Chow polynomials: for Gorenstein* posets they are unimodal, and for face posets of polytopes they are γ-positive. Along the way we provide negative answers to two open questions concerning Eulerian and Gorenstein* posets. First, the question on the nonnegativity of Eulerian Chow polynomials, posed by Ferroni, Matherne, and Vecchi (2024). Second, the question posed by Athanasiadis and Kalampogia-Evangelinou (2023) on the real-rootedness of chain and Chow polynomials of Gorenstein* posets: these examples provide a novel application of a technique introduced by Murai and Nevo (2014).
ER -