The equivariant Ehrhart theory of polytopes with order-two symmetries: Part II
Abstract
We give two counterexamples to Stapledon's Effectiveness Conjecture. Each counterexample is a $6$-dimensional lattice polytope with an order-two symmetry whose equivariant H*-series φ[t] is a polynomial with one coefficient that is not effective. For polytopes with a symmetry of order two, the goal of this paper is to build up a picture of the landscape of results for when the Effectiveness Conjecture holds. Our main result is a criterion, in terms of the faces of P, that determines when the Ehrhart ring of P admits an invariant linear system of parameters of the correct representation type. We use these results to reduce the space of possible counterexamples to one of two cases. A randomised computer search produced counterexamples in each case. We also give an example where the equivariant H*-series is effective but admits no invariant linear system of parameters, neither for P nor any invariant subdivision of it, answering a question of Stapledon in the case of groups of order two.
Related
- Equivariant Ehrhart series for rational simplices and rational cross-polytopes
- Ehrhart polynomials of cyclic polytopes as averages of zonotope Ehrhart polynomials
- Ehrhart positivity of smooth polytopes with few facets
- Ehrhart Properties under Operations on Lattice Polytopes
- Exact Ehrhart Series of Birkhoff Polytopes via Constant Terms and Finite-Field Evaluation
BibTeX
@misc{clarke2026equivariant,
title = {{The equivariant Ehrhart theory of polytopes with order-two symmetries: Part II}},
author = {Oliver Clarke},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17988},
eprint = {2610.17988},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17988},
}
RIS
TY - UNPB
TI - The equivariant Ehrhart theory of polytopes with order-two symmetries: Part II
AU - Oliver Clarke
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.17988
UR - https://arxiv.org/abs/2610.17988
AB - We give two counterexamples to Stapledon's Effectiveness Conjecture. Each counterexample is a $6$-dimensional lattice polytope with an order-two symmetry whose equivariant H^*-series φ[t] is a polynomial with one coefficient that is not effective. For polytopes with a symmetry of order two, the goal of this paper is to build up a picture of the landscape of results for when the Effectiveness Conjecture holds. Our main result is a criterion, in terms of the faces of P, that determines when the Ehrhart ring of P admits an invariant linear system of parameters of the correct representation type. We use these results to reduce the space of possible counterexamples to one of two cases. A randomised computer search produced counterexamples in each case. We also give an example where the equivariant H^*-series is effective but admits no invariant linear system of parameters, neither for P nor any invariant subdivision of it, answering a question of Stapledon in the case of groups of order two.
ER -