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arXiv · 2610.17988

The equivariant Ehrhart theory of polytopes with order-two symmetries: Part II

Oliver Clarke
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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.

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Oliver Clarke. 2026-10-13. The equivariant Ehrhart theory of polytopes with order-two symmetries: Part II. https://arxiv.org/abs/2610.17988

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